The 1962 Cuban Missile Crisis is conventionally seen as one of, if not the, closest time the US and Soviet Union came to nuclear war. There’s tons of literature available online on the topic if you’re interested.

One more obscure event you may be interested in, however, was an incident in 1983 when the Soviet Union detected the launch of multiple USAF Minuteman ICBM’s towards the USSR. A Soviet Air Defense Forces officer named Stanislav Yevgrafovich Petrov realized that the strike was a false alarm (he believed his radar system was not functioning properly) and prevented the Soviets from launching retaliatory missiles which likely would have ended the world as we know it (once the USA responded to what it would have perceived as an unprovoked nuclear attack). Very interesting, and often not widely discussed, episode in Cold War history.