Multi-User Dungeons, or MUDs, have been around since the late 1970's, but only gained wide popularity in the early 1990's with the release of several public codebases. A codebase is the engine on which a game world exists, dictating the kinds of things allowed in the world, the kinds of actions things may take, the limits of their power, and so on. A handful of codebases established themselves in the late 80's and early 90's, and from those few codebases numerous other kinds of MUDs evolved, some of them advancing far beyond their predecessors.
The Wikipedia has a good history of MUDs, including a list of the more popular codebases. The majority of codebases today derive from the DIKU codebase, including Circle, Merc, ROM, and Smaug, and from those codebases programmers developed tbaMUD, GodWars, EmberMUD, AFKMud and countless others. While DIKU was certainly the most popular codebase to work with, other early codebases included AberMUD (accepted as the first widely popular MUD), TinyMUD and LPMud, as well as the closed-source game GemStone. It's hard to find a MUD today whose origins do not trace back to those early codebases.*
Your favorite search engine will reveal where you can download these various codebases. However, I'll point out what I think is the most prominent repository, and link to a few sites who offer a home either to a finished codebase or to a codebase with active development.
MUDBytes - MUDBytes hosts forums, articles, MUD lists and more. It also serves as a repository for codebases, code snippets, bug fixes, areas, helps and anything else related to text-based gaming. As an added bonus, the site breaks down the codebases by their derivatives; DIKU, as mentioned above, is far and away the most popular parent of them all.
The Builder Academy (tbaMUD) - CircleMUD, a DIKU derivative, stopped active development in 2006. The Builder Academy, a game based on CircleMUD, continued to refine its own code, and in that same year released its codebase as tbaMUD. The administrators there published version 3.65 of tbaMUD in February of 2014. You can download the latest version of tbaMUD from their website or from GitHub.
The CircleMUD Home Page still functions if you're inclined to learn a little about it.
SmaugMuds.org - This site offers several codebases derived from the Smaug codebase. DIKU and most of its derivatives are based on the C programming language; the Smaug derivative AFKMud (offered at SmaugMuds.org) switched over to C++. SmaugFUSS, another offering from SmaugMuds.org, is an attempt to remove as many of the bugs found in the original Smaug as possible. SmaugFUSS was last updated in October of 2014 and can be grabbed from GitHub.
By the way, if you're interested in the history of this game, the site offers an unofficial hierarchy of Smaug and its derivatives.
LDMud - While DIKU is the most popular codebase as far as MUDs go, there are other codebases publicly available and in active development. LDMud, for example, is the continuation of the original LPMud, which ended development in the 1990's. The last stable release of LDMud came out in October, 2017.
* But not impossible. There are several closed-source codebases in existence today that people play. The Interactive Fiction Engine by Simultronics is one, as is the Rapture Engine by Iron Realms Entertainment, and there's the codebase behind the Eternal City by Skotos Tech. Godwars II, while it shares the same name with GodWars, is a newer (and unreleased) codebase altogether.
Grim Wheel is a site dedicated to old school text gaming.
Click the links below to go from section to section of this site.
Buildcraft - Articles which discuss the craft of building on Multi-User Dungeons (MUDs).
MUD Forums - Links to offsite forums that concern themselves with all things related to Multi-User Dungeons.
MUD-Dev - Where you can find archives of the former mailing list known as "MUD-Dev", as well as a link to the newest mailing list.
Imaginary Realities - Archives of articles from the now-defunct online magazine Imaginary Realities.
MUD Listings - Websites which list the many text games still in existence.
MUD Clients - Clients which you can use to connect to an active MUD.
Netbooks - A repository of supplements for tabletop roleplaying games.
Miscellaneous - Links that do not fall easily into any other category.
Codebases - The various public codebases in existence today.
Area Archives - Areas for public codebases such as Circle, ROM and Smaug.