What event made Americans fearful of a nuclear attack by the Soviets?

Americans had reason to fear a Soviet nuclear attack even without any specific event, since the US and the USSR were clearly established as ideological and political enemies, from 1945 until the fall of communism in 1990, however, there was one particular event which was more frightening than other events of the Cold War, and that was the Cuban Missile Crisis of 1962. That was the closest we ever came to an actual nuclear war.