X-COM Bug FAQ
X-COM: UFO Defense (DOS versions 1.2, 1.4, and Windows Collector's Edition)
X-COM: Terror From the Deep (DOS versions 1.0, 1.2, and Windows Collector's Edition)
X-COM: Apocalypse (DOS 1.0, Windows Collector's Edition)
Please note that this information is all coming from my personal experience in dealing with the thousands of people who have emailed me with their problems and therefore contains the extent of my knowledge on getting the xcom games to run successfully. After you have tried everything listed in the FAQs, consider the following options:
If the xcom game just won't work, boot to MS-DOS mode and try it (assuming you're running the DOS version of the game).
If it still doesn't work, reinstall the game.
If it continues to refuse to work, reinstall Windows.
If it still just won't work at all or continuously crashes, install a different version of Windows.
Anything beyond this has got to be hardware problems. Pick up an old 486 from the newspaper classifieds for $25.
However, the solutions that are in the FAQs as of the writing of this page lead to a 95% success rate among xcom fans, so chances are good that you will get your game working.