Non-violent Games

Non-violent games (designed for players to achieve in-game goals using completely non-violent strategies) might be more abundantly available now than they ever have been in the past, even if they tend not to garner as much mainstream media attention as violent ones:
Unlike in the past, when the majority of nonviolent games were found in casual or sports-related games, non-violent games are increasingly easy to find in many other genres, notably story-based adventure games and "simulation" games, as well as a minority of RPG games, among others. Part of the growth in non-violent gaming is due to the popularity of casual gaming markets in recent years, but successful non-violent titles involving more complex engines and demanding gameplay are also becoming more common.
Awakening: Moonfell Wood, one of many non-violent adventure games released within the past decade.
Interestingly, until the rise of World of Warcraft (in which "pacifist challenge" gameplay is reportedly optional, but definitely not the norm), the two best-selling PC games of all time were both non-violent: Myst (released in 1993) and later The Sims (released in the year 2000), both proving that non-violent games could be just as commercially successful as violent counterparts, or even more successful.
Myst stood out from most other games in the early 1990s for having no violent gameplay, yet surprised many by becoming the best-selling PC game of all time for many years.
Most of the early computer RPG games were based on a dungeon-crawl model that involved mandatory combat, but the same was not true of the adventure genre, which has usually tended to be less oriented toward violent conflict: 1976's text-based Colossal Cave Adventure was oriented around treasure-hunting and exploring, 1984's Below the Root was mostly devoid of violence, and the influential 1980s-90s King's Quest series was oriented toward exploring, solving puzzles, and the unfolding of a faery-tale storyline. Deadly combat encounters in adventure games were reasonably rare (compared with their counterparts in RPG and other genres) and were often over relatively quickly when they took place, unlike genres in which "grinding" by repetitively smiting monsters and foes was the normal mode of gameplay.
King's Quest (shown in the AGD Interactive remake with updated graphics) involved brief moments in which Sir Graham had to confront monsters such as an evil witch, a giant, and a dragon, among others. However, the mood and mode of gameplay in point-and-click adventures like King's Quest was mostly non-violent.
Adventure games by Sierra, LucasArts, and other developers were very popular for a period of time, and defined a new popular genre in PC gaming, usually based on "point-and-click" gameplay with a low learning curve, relying on solving mysteries and puzzles to create a challenge for players. The low learning curves only made these games easy to learn to play, however, and not necessarily easy to win: Many early adventure games were notable for requiring the player to solve extremely challenging and obtuse puzzles, some of which have been compiled in lists of "most aggravating" or devious puzzles ever encountered in adventure games.
Classic adventure games like King's Quest were more likely to challenge players with extremely difficult puzzles, rather than by forcing players to engage in surfeits of combat. Shown above is the notorious gnome who required Sir Graham to guess his name...
In 1993, Cyan released a unique game called Myst that was built on a somewhat similar framework, using first-person perspective travel via a point-and-click interface through a world rendered in still images, with occasional animations. The game placed the player alone on a mysterious island with many mysterious devices whose functions were not readily apparent, requiring them to gradually explore, gather clues, and figure out how to progress using logic and puzzle-solving skills. Despite the simplicity of its game engine, Myst was so absorbing to thousands of players that it became the best-selling PC game of all time, holding its record all the way into the year 2000 (when "The Sims" was released).
The first area on the mysterious island the player must explore, from the original release of Cyan's Myst.
Later in the 1990s, however, adventure games fell strongly out of favor with commercial developers for various reasons, and by some accounts became near-"untouchables" with mainstream production companies for a period of time, leading to many games by once-popular designers being scrapped while still in production. Fortunately, many adventure game designers (including some who had lost their jobs at former adventure game developers like LucasArts) found new ways to get back into the markets without the backing of big-name companies, eventually forming new companies and using new online distribution platforms to re-introduce new adventures. These included both adventure games released into "casual" gaming markets, and higher-budget titles with a somewhat cinematic approach to storytelling.

One of the most abundant sources of completely non-violent games are the PC adventure games developed by companies like Big Fish (and various independent developers releasing games through Big Fish and other distributors). Most of these feature a story-based game with remarkably beautiful painted and/or 3d-rendered landscapes, which are further brought to life using animated visual and lighting effects. In most cases, players navigate through the in-game world using a simple point-and-click interface following the traditions established by earlier generations of PC adventure games on MS-DOS and other operating systems.
Dreampath: Two Kingdoms, another non-violent game from the new generation of adventures that combines the accessibility level of "casual" games with the classical point-and-click exploring and puzzle-solving model of the DOS classics from 1980s and 1990s.
Like many of the popular classic era adventure games, the new generation of games are often developed by re-using similar (or identical) game engines for large numbers of releases that feature new artwork and storylines, enabling designers to be much more prolific and release more games at lower costs than was once the case, back when the adventure genre fell out of favor with larger companies in the late 1990s. Most of the newer wave of adventure games give players the option of playing on either a "casual" mode of difficulty that makes the game fairly easy, while players in search of a challenge can select at least one harder difficulty mode that removes clues, penalizes player error in some cases, and/or imposes time limits on players during attempts to solve puzzles.
The setting and style of these games tends to vary fairly widely: beautifully realized fantasy-themed settings are common and popular in dozens of games, many featuring very positive atmospheres, cute animal companions, and completely non-violent storylines. However, dozens of others have much darker themes with mystery, occult, or horror settings, and some feature sinister imagery or themes that some parents might find objectionable. However, even in the darker games, using violence to overcome obstacles appears to be a rarity, with even the horror-themed games more often requiring non-violent solutions to the conflicts and dangerous encounters found in the game's storyline.
A cave in Detective Quest: Crystal Slipper, which features a few more macabre fairy-tale themes and visual elements, but is still arguably on the benign side. Dozens of other recent adventure games modeled on the same engines are much darker and overtly horror-themed, but still rarely appear to involve requiring violent actions on the part of the player.
Nonviolent Simulations

SimCity may take the prize for being the longest-running and most popular non-violent simulation/strategy game series. Since the first release in 1989, SimCity became famous for its ability to absorb the attention and energy of players preoccupied with building and managing their own cities, despite the moderately high learning-curve and level of complexity found in all releases. Numerous other non-violent "Sim" games have been released since the original 1989 SimCity release, meeting with varying degrees of success, but in the year 2000 the "life-simulator" called The Sims became the best-selling PC game since Myst was released.
A coal and ore mining city under development in SimCity 2013, the latest sequel since the original SimCity was released in 1989. The original 1989 SimCity was only released after years of difficulty getting distributors to take a risk on the open-ended and non-violent city building game, which was unusual for its time.
Many new non-violent entrants to the simulation genre have been appearing in what some consider the "casual" games marketplace as well: Examples include Animal Crossing and Nintendogs (both produced by Nintendo), Farmville and other non-violent social-media games, and dozens of simulations at Big Fish Games which feature non-violent themes like managing a tribe of castaways, or wealth-building or farming simulation games. Many are playable on mobile devices, and some popular entries are integrated with social networks.
Cover art for Animal Crossing by Nintendo

Nonviolent Strategy Games

Non-violent strategy games on par with the most successful titles in the war-strategy genres (for example, Warcraft, Command & Conquer, Age of Wonders, or Heroes of Might & Magic) so far appear to be fairly rare, and we're not aware of major game developers that have tried to create a completely non-violent turn-based or real-time strategy game similar to the fantasy epics like the Warcraft (in real-time) or Heroes of Might & Magic series (turn-based).
Age of Wonders is one of many absorbing turn-based strategy games that revolves around warfare and battles, though in some cases user-created maps and campaigns can be created in such games to define non-violent objectives.
Many of the best real-time and turn-based strategy games actually came with editors that allowed users to create their own maps and "mods" that would theoretically make designing non-violent campaigns possible. A few possible examples include Warcraft II (which enabled players to create their own maps, and also to set non-violent victory conditions), and the high-quality open-source strategy/RPG game Battle for Wesnoth, which has dozens of extensive user-created campaigns with extensive storylines, including adaptions designed to play like RPGs.

One "crossover" game representing a step in the direction of non-violent strategy/RPG games might be the non-violent Viking Saga series: 
The Viking Saga series and many other non-violent games available through Big Fish Games (among other places) blur the line on their exact genre, and can be found listed as "Adventure," "Time-Management," "Strategy," or all of the above.
The series is part of the "casual" game market designed to have a low learning-curve, and so can't be directly compared to classic series like Heroes of Might & Magic, Warcraft, or Age of Wonders. However, the Viking Saga games still represent a successful blending of real-time strategic gameplay with a detailed storyline comparable to that found in some of the more light-hearted RPGs. The first game is a lengthy plot-driven quest about a young Viking's quest to sail to a distant land and dispose of his father's cursed ring. Lots of humorous interaction and dialogue with "NPC" characters takes place between stages, combined with other fantasy-RPG elements like exploring one's way through a world map one stage at a time. However, the gameplay is based on strategically sending the mettlesome vikings out to perform various tasks like clearing away boulders, gathering resources, and building structures to complete quest-specific goals.
The first challenge in Viking Saga is to gather food, prepare a small shelter, and repair a dock after a shipwreck , in order to recover the hero Ingold's treasure chest. The closest the quest comes to involving combat is to "drive away" creatures like giant spiders found dwelling inside a series of caverns.
With no foes to slay, the challenge in Viking Saga comes primarily from having to build structures, gather and manage resources, and complete goals within a limited time in each stage, requiring the player to use strategy to decide which order to complete tasks in. Like most Big Fish games marketed toward "casual" gamers, however, players have the option to play on either a hard difficulty level setting that demands an optimized strategy, or a "casual" setting that gives players more time to complete goals at a more relaxed pace, and focus on enjoying the unfolding story as the vikings bloodlessly forge their way beyond a storm-wracked coastline, though swamps, forests, mountains, and caves.

Non-violent Racing Games & Sports Games

Sports games have long been popular, and  1958's "Tennis for Two" has been cited as the first video game ever created, while "Pong" became a huge hit that launched Atari's initial success into mainstream gaming. Today there are more high-definition sports and racing titles than ever, many of which involve no violence. To capitalize on the popularity of sports, Nintendo sold its Wii console with the game "Wii Sports" as bundled software to all regions other than Japan.
Racing games also have a long and popular history, and are popular with many gamers who have no interest in other types of sports games. The racing genre has long included "battle racers" that were sometimes overtly violent, but non-violent racing games have always been popular as well.
Super Mario Kart on the Super Nintendo
Some of the landmark titles that popularized racing games were Sega's Out-Run, Nintendo's futuristic F-Zero and Mario-themed "crossover" Super Mario Kart on the SNES. The games inspired both sequels and imitations. Both "arcade"-styled racers and more realistic racing simulation games like the Gran Turismo series and Forza series also started to become growingly popular as newer generations of console raised the standards in realism in graphic definition and simulated physics, among other things.

Non-violent Experimental Games

Independent game developers have been releasing a growing number of experimental non-violent titles that often have a presentation that makes their games look more like art projects than standard commercial games, but some have been finding commercial success, as well as cult followings and critical acclaim. A few examples include Gone Home, Dear Esther, Flower (a game in which the player enters the dreams of flowers and controls the wind), and a non-violent exploration-based multiplayer game called "Journey" in which players communicate via a musical chime and do not have the option of attacking each other or flaming each other using text or speech. Many of these games have very impressive visuals and production values, and some were very successful commercially despite breaking dramatically from mainstream trends.
Screenshot from the nonviolent PS3 title "Flower"

Non-violent RPGs: Past and Future Prospects

Traditional pen-and-paper RPG systems (most famously the Dungeons & Dragons systems) have a tradition of giving players wide latitude to create their own characters and choose their own inventive strategies for completing quests, including role-playing completely non-violent characters. The freedom to create a character of choice and solve quests using non-violent strategies has been implemented impressively in many otherwise violent RPGs, including games in the Elder Scrolls, Two Worlds, Eschalon, and Fallout series, to name a few. However, RPGs that are inherently non-violent and do not involve violence as a central theme and gameplay mechanism still appear to be something of a rarity.
Even RPGs as uplifting as the SNES classic Chrono Trigger almost always require the player to engage in hundreds of bouts of combat with various enemies, as when these three  "monsters" attack Chrono in the otherwise idyllic setting of Guardia Forest.
Up until recently, the only completely non-violent RPG series I was aware of was Harvest Moon (later renamed "Story of Seasons"), an unusual 1996 SNES console RPG centered around restoring an old farm using non-violent strategies. The game had many sequels on multiple platforms, and was very influential on highly popular farm-simulation "casual" games like Happy Farm and Farmville.
Natsume's Harvest Moon: Like in many console RPG intros, a snowcapped mountain range rises above green fields and evergreen forests, beckoning the hero onward toward adventure— but in this case, the quest is to restore an old farm and become a prosperous, non-violent young farmer...
Harvest Moon's legacy proved that an RPG game could be successful and influential without involving combat, even though its light-hearted setting was admittedly somewhat less "epic" than its counterparts in the Final Fantasy, Dragon Quest, Breath of Fire, or other "golden age" series of console RPGs released in the 1990s. Some of the reputed non-violent RPG games we've heard of haven't been released outside Japan (including many interactive novels that have only recently started being translated for overseas markets).
The industrious hero of Harvest Moon is shown hard at work on his farm, under the observation of an animal companion.
One notable new offering by an independent developer is the new console-styled RPG "Undertale," sometimes seen with the tagline "The RPG game where you don't have to destroy anyone." Undertale features mandatory combat encounters, but provides the ability to spare the lives of the monsters the player encounters in the game, and directly weaves the theme of avoiding killing into the game's plotline. Undertale very quickly became a cult success, proving that "pacifist run" concepts are no obstacle to an RPG game being successful.
Screenshot from Undertale
Our guess is that it's only a matter of time before we start seeing many more completely non-violent RPGs: In addition to diehard RPG fans who were never strongly attached to turn-based (or other) RPG combat to begin with, the casual games marketplace has also brought in thousands of new gamers as a "gateway" that is converting non-violent casual adventure game players into RPG players as well.
Aveyond II: Ean's Quest isn't a pacifist run RPG, but is an example of a game we've noticed being well received by casual gamers who were introduced to the Aveyond series as their first RPG'ing experience.
Big Fish Games and other distributors whose libraries include independent RPG series titles like Aveyond, Laxius Force (and its prequel 3 Stars of Destiny), Skyborn, and other independently developed RPG-Maker titles have been putting console-styled RPGs in front of PC game players who might otherwise never have heard of either the individual games, or the RPG genre itself. Many of these titles have met with lots of rave reviews from former casual gameplayers (some of whom stated that these were the first RPGs they ever played or encountered) who are now happily buying and playing RPGs, as well as non-violent casual games and non-violent adventure titles.
Most of the new "old-fashioned" console-styled RPG games we've discovered still involve a significant amount of mandatory combat, however. One of the reasons for this is simply the RPG tradition, which features legendary stories involving battles. Another reason is that many newer console-styled RPGs appear to have been created using an English-language edition of the RPG-Maker software, which features a classic turn-based combat engine by default.
Aveyond II
However, since many gamers discovering RPGs for the first time on sites like Big Fish are already part of a market of buyers who avidly buy and play completely non-violent adventure games, the door has obviously been opened for designers to develop completely non-violent RPGs as well— whether they rely on RPG Maker, or other development tools. The commercial success of other non-violent game genres probably indicates that it's only a matter of time before lots more non-violent RPGs start being developed and released.