1) Cuban Missile Crisis- When the US put missiles in Turkey, the Soviet Union responded by putting missiles in Cuba. This event, that took place during October 1962, was the closest the world has come to nuclear warfare. The 13 days in October in which the Soviet Union and US were in intense diplomatic talks were some of the most tense days in world history.
2) Development of North Atlantic Treaty Organization (NATO)- This political and military alliance between the US and several European countries (like Great Britain, France, and Denmark) resulted in the Soviet Union developing the Warsaw Pact (which was also a political and military alliance). The forming of these alliances appeared as a means to prepare for another war by the US and Soviet Union.
3) Kitchen Debate- This impromptu debate between Richard Nixon and Nikita Khrushchev over the structure of the US and Soviet economy caused increased tensions between the two countries.