The International Space Station has fostered the cooperation of five space agencies, including those of the US and Russia. While well known today, the ISS was not the first time the US and Russians cooperated in space.
Apollo–Soyuz was the first crewed international space mission, carried out jointly by the United States and the Soviet Union in July 1975. Millions of people around the world watched on television as an American Apollo spacecraft docked with a Soviet Soyuz capsule. The project, and its handshake in space, was a symbol of détente between the two superpowers during the Cold War.
The mission just had it’s 49th anniversary yesterday.
The technical legacy continues until this day:
A derivative (but mechanically incompatible) docking collar, APAS-89, was launched as part of the Kristall module of the Soviet Mir space station. Originally intended as the docking port for the (defunct) Buran Soviet space shuttle, the APAS-89 unit was used for the next Russian-American docking mission, STS-71, twenty years later as part of the Shuttle–Mir program (though not before the docking port was tested by the last APAS-equipped Soyuz, Soyuz TM-16, in 1993).
The American Space Shuttle continued to use the same APAS-89 docking hardware through the end of the Space Shuttle program to dock to Mir and then the International Space Station, the latter through the Pressurized Mating Adapters (PMAs).
You can read more about Apollo-Soyuz on Wikipedia.