Throughout the Cold War, there were several times when the United States and the Soviet Union nearly went to war. A policy known as brinkmanship meant that both powers frequently pushed the envelope of foreign policy to the point where war would be likely if one side did not back down. It is thought that only the promise of mutually assured destruction by nuclear weapons prevented an all-out conflict.
Perhaps the hottest moment in the Cold War was the Cuban Missile Crisis in October of 1962. In an attempt to match US deployments of nuclear weapons in Italy and Turkey, the Soviet Union stationed nuclear missiles in Cuba. This led to a showdown as the US Navy blockaded Cuba. There were several tense moments when it looked like actual fighting would take place. In the end, diplomacy prevailed, and both nations agreed to recall many of their nuclear weapons from overseas.
Another hot moment in the Cold War was the Berlin Blockade. From June 1948 to May 1949, the Soviet Union, in an attempt to limit access to Berlin by western powers, cut off access to West Berlin. This threatened to cut off essential supplies to West Berlin and possibly force it to capitulate to the Soviets. The United States, Great Britain, and France responded with the Berlin Airlift, in which they flew essential supplies to the city, bypassing the blockade.
On May 1, 1960, an American U-2 spy plane was shot down over the Soviet Union. CIA pilot Francis Gary Powers successfully ejected from the plane but was captured by the Soviets. This was an incredibly embarrassing incident for the United States, as it proved that they were spying on their Cold War rival and violating their sovereign airspace. This occurred just a couple of weeks before a scheduled peace summit in Paris. When Eisenhower refused to apologize, Khrushchev walked out of the talk, and the already tenuous relationship between the US and the USSR became even worse.