Historically, MUDs (and their more recent progeny with names of MU- form) derive from a hack by Richard Bartle and Roy Trubshaw on the University of Essex's DEC-10 in the early 1980s; descendants of that game still exist today and are sometimes generically called BartleMUDs. There is a widespread myth (repeated, unfortunately, by earlier versions of this lexicon) that the name MUD was trademarked to the commercial MUD run by Bartle on British Telecom (the motto: "You haven't *lived* 'til you've *died* on MUD!"); however, this is false -- Richard Bartle explicitly placed `MUD' in the public domain in 1985. BT was upset at this, as they had already printed trademark claims on some maps and posters, which were released and created the myth.
Students on the European academic networks quickly improved on the MUD concept, spawning several new MUDs (VAXMUD, AberMUD, LPMUD). Many of these had associated bulletin-board systems for social interaction. Because these had an image as `research' they often survived administrative hostility to BBSs in general. This, together with the fact that Usenet feeds were often spotty and difficult to get in the U.K., made the MUDs major foci of hackish social interaction there.
AberMUD and other variants crossed the Atlantic around 1988 and quickly gained popularity in the U.S.; they became nuclei for large hacker communities with only loose ties to traditional hackerdom (some observers see parallels with the growth of Usenet in the early 1980s). The second wave of MUDs (TinyMUD and variants) tended to emphasize social interaction, puzzles, and cooperative world-building as opposed to combat and competition. By 1991, over 50% of MUD sites were of a third major variety, LPMUD, which synthesizes the combat/puzzle aspects of AberMUD and older systems with the extensibility of TinyMud. In 1996 the cutting edge of the technology is Pavel Curtis's MOO, even more extensible using a built-in object-oriented language. The trend toward greater programmability and flexibility will doubtless continue.
The state of the art in MUD design is still moving very rapidly, with new simulation designs appearing (seemingly) every month. Around 1991 there was an unsuccessful movement to deprecate the term MUD itself, as newer designs exhibit an exploding variety of names corresponding to the different simulation styles being explored. It survived. See also bonk/oif, FOD, link-dead, mudhead, talk mode.