Of late such blogging tools and sites as Live Journal have been utilized for this purpose.
This includes such games as play-by-email (or PBEM) RPGs.
While these are often seen as definitive boundaries, exceptions abound; many MUSHes have a software-supported combat system, while a "Role-Playing Intensive MUD" movement occurred primarily in the Diku MUD world, and both the first Internet talker (a type of purely social server) and the very popular talker software ew-too were based on LPMud code.
Rather than following gameplay in real-time, such as in MUDs, players post messages on such media as bulletin boards, online forums, Chatrooms (such as like AOL,hangouts and Yahoo chat) and mailing lists to which their fellow players will post role-played responses without a real limit or timeframe.
The origins of this style of role-playing are unknown, but it most likely originated in some form during the mid-to-late 1980s when BBS systems began gaining in popularity.
Usually it is played through 'Script' and 'Story' format, both styles are interchangeable and work well but it depends on which the player prefers, or which the human administrator insists upon.
Script format is a simple stating of what each character is saying, post by post, with little to no mention of said characters' actions, whereas Story format requires that the character's actions be mentioned, including the surroundings and a general description of what is going on.
Some games rely entirely upon human moderators to dictate events, and physical print books for rules sets.