You're right treadstone71... the game provided a very extensive background, covering from 1995 to 2000 (In the game timeline) that trigger the WWIII. Too long to explain here but i'll try to make it short...
(From Wikipedia)
1st Edition
In 1995 a series of Sino-Soviet border conflicts expanded into general war between the Soviet Union and China. The Sino-Soviet war rapidly escalated from conventional warfare into exchanges of nuclear, chemical, and biological weapons.
In 1996 a cabal of East German and West German military officers seek to reunify their country. Unified German forces stage a coup d'état against the Group of Soviet Forces in Germany. The US and NATO allies initially attempt to stay out of the war, but are quickly drawn into the European conflict.
Between 1996 and 1997 a largely conventional war is fought between NATO and Warsaw Pact forces throughout Europe. There are limited exchanges of battlefield nuclear weapons, and chemical and biological weapons.
During Thanksgiving 1997, the Soviet Union launches a surprise first strike against targets in the United States and Europe. The US and Great Britain launch retaliatory nuclear strikes against the Soviet Union.
In the aftermath of the nuclear exchanges, both blocs struggle to recover from the damage. The war continues - despite increasing shortages of men, equipment, and fuel. In the United States there is a breach between the civilian government and the Joint Chiefs of Staff. An open rift develops between "Civgov" (the civilian government) and "Milgov (the military government), which leads to a low-intensity civil war.
By the summer of 2000, the European theater of operations had been fought to a near stalemate. In one final effort to break the deadlock and end the war, NATO forces planned a summer offensive across Northern Poland and into the Baltics. The offensive ground to a halt in the face of a Warsaw Pact counterattack. Several divisions & corps on each side were virtually eliminated. In the chaotic aftermath, some small groups of survivors sought to find their way “home’. Others turned into gangs of marauding bandits. Others went “native’ and integrated with the militias of independent “free cities".
This is the universe in which characters operate.
Second Edition
The second edition, published in 1990, featured a reworked background intended to be an alternate history closer to 1990s real-world history. Version 2.2, the game's final edition, was published in 1993 and featured a background in which the KGB's Alpha Group obeyed the coup leaders in the August 1991 Soviet coup attempt and stormed the Russian White House, killing Boris Yeltsin and effectively preserving communist control.
The scenarios in Twilight 2000 typically involve a military unit which was stranded in Central Europe after the nuclear war and places emphasis on attempting to realistically depict military and social systems after a nuclear war.
Twilight 2000's success as a game can be attributed in part to its manner of presenting a military background and setting, without hemming the players into a military's command structure. The civilian governments of most countries in the game have been shattered (the only countries still mostly intact in Europe are France and Belgium, which refused to enter into the war, leaving NATO) with the US government splitting into the civilian and military governments, as has the military chain of command, the supply lines, etc. The various military forces are presented as being much weaker than their stated sizes, occupying civilian territories and relying on civilians for food and small-scale manufacturing, and recruiting from civilian populations to some extent. This all means that the players can feel they are part of a military of some sort, without their characters being forced to submit to higher military authority. Players generally can choose whether they want their players to try and continue the war, get back home (wherever that may be), be a part of one of the new power groups wherever they are, simply survive as mercenaries or marauders, or some combination.
In 1990, realizing that the game's background was in danger of become obsolete, GDW published Merc 2000, an alternative campaign set which revolved around mercenaries fighting brushfire wars in a time in which the Twilight War never occurred. After Merc: 2000 was released, many supplements and articles printed in GDW's Challenge Magazine featured Twilight 2000 with equipment and background conversions to Merc 2000 or were Merc 2000-only.
In 1991 GDW licensees Paragon developed a computer game adaptation (complete with expansion, "the Colonel") of Twilight 2000 depicting a squad of 20 soldiers stranded behind enemy lines in Poland, struggling against the despot Baron Czarny.
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Find more information here:
http://en.wikipedia.org/wiki/Twilight_2000