The competition between the United States and the Soviet Union for superiority in extraterrestrial exploration was called Space Race.
America had reason to compete against the Soviet Union in Space Race: Sputnik beat the United States to space in 1957, damaging its national pride and igniting fears of Soviet technology. The Soviet Union wanted space just as badly as the United States did. The Soviet Union and the United States differed in how they propagandized the Space Race, but they shared the common bond of actually having space programs.