During the Cold war, there were constant threats and tensions, but never a direct confrontation between the two opposing superpowers: the US and the URSS, which leaded the Western capitalist bloc and the Eastern communist bloc respectively.
Although both countries had access to nuclear weapons, those from the US were even more sophisticated. This is why president Eisenhower could use the threat of massive retaliation to prevent attacks. He claimed that the US would respond to any attack by retaliating in much greater force. Such attacks, involving nuclear weapons, would led to destruction of the two nations and the whole world. This is why a direct attack was avoided and, instead, the two powers participated in proxy wars as opponents, such as in the Korean War or the Vietnam War.