Conflicts between the U.S. and USSR and their proxies were interspersed with periods of relatively less conflict (détente) and even cooperation between the two powers. In the 1956 Suez crisis, when Britain, France and Israel attempted to retake the Suez Canal from Egypt, both the U.S. and the USSR opposed it. On a system level, this can be seen as the two newly-dominant rivals cooperating to prevent the old imperial powers of Britain and France from re-establishing their power.
Other cooperation included nuclear arms control treaties such the 1963 Atmospheric Test Ban Treaty, the 1968 Nonproliferation Treaty and the 1972 SALT I and 1979 SALT II nuclear arms limitation treaties. There was also some trade between the two countries.
By the 1980s, discontent was growing within the USSR and the economy was stagnating. In 1985, a new factor appeared in the person of Mikhail Gorbachev as a reformist leader. He withdrew the USSR from its disastrous war in Afghanistan, allowed Eastern Europe countries to become independent, signed the START arms reduction agreements with the U.S. and tried to reform the USSR through perestroika (restructuring) and glasnost (openness). Capitalist West Germany and Communist East Germany, split by the Cold War, were reunited and the USSR’s Warsaw pact in Eastern Europe fell apart. The Berlin Wall was destroyed by protestors. After a failed 1991 coup against Gorbachev by Soviet reactionaries, the USSR broke up into different countries, and the Cold War was over.