At the moment, just about every computer game published has its own code for rendering graphics, simulating physics and so on. Sometimes this code is at least partially reused from game to game (e.g. Source), but each game still comes with its own tuned and updated version of it.
I think at some point the games industry will reach the point where game engines are independent of the games themselves. In other words, there will be a common "language" that game designers will use to specify how the game will work and to store assets (graphics, models, sound, music etc.) and there will be multiple client programs that can interpret this data and be used to actually play the game. Some engines will naturally be more advanced than others - these may be able to give extra realism to games not specifically written to take advantage of it. And games written for later engines may be able to run on earlier ones with some features switched off.
Many classes of software have evolved this way. For example, in the 80s and early 90s there were many different ways of having rich documents and many such documents came with their own proprietary readers. Nowadays everybody just uses HTML and the documents are almost always independent of the browsers. As far as 2D games are concerned, this convergence is already happening to some extent with flash.
3D games have always pushed hardware to its limits, so the overhead of having a game engine not tuned for a particular game has always been unacceptable. But as computer power increases, this overhead vanishes. Also, game engines are becoming more difficult to write (since there is so much technology involved in making realistic real-time images) so there are economies of scale in having a common engine. Finally, I think people will increasingly expect games to be multi-platform, which is most easily done if games are written in a portable way.
If game design does go this way, I think it will be a positive thing for the games industry - it will mean that more of the resources of the game can be devoted to art, music and story-telling. This may in turn open up whole new audiences for games.
[...] was thinking some more about convergence of game engines recently, and started wondering what a cross between a web browser and a game engine would look [...]