During the war, there are usually physical fighting between warring countries. However, during this period of history, the United States and the armed forces of the Soviet Union has never met on the battlefield.
The two countries have been constantly react to the actions of the other though. This included espionage, a military weapons competition, industrial and technological developments, and even the moon race. The two countries have spent vast sums of money on defense, nuclear weapons, and support for third wars.
Because both countries are beaten them directly, it was called a “cold” war means that there was no physical fight.
Opponents of the Cold War often provide economic or military aid, such as weapons, tactical support and military advisers, to nations less involved in conflicts with the opposing country.
There were a number of armed conflicts with the two main players on opposite sides – the Korean War, the Vietnam War, the Soviet war in Afghanistan. Arab Israeli wars. Various comes on in Africa. The Indo-Pakistani War.
There was also the Cuban Missile Crisis of which failed in the US and actually shot the USSR to another.
So yes, there was a lot of fighting. But because of nuclear deterrence on both sides have been careful to avoid direct confrontation, fighting by proxy. The United States to suffer about 140,000 victims of the Korean War and about 360 000 injured in the Vietnam war. The Soviets suffered 68,000 casualties in Afghanistan. These figures, although overshadowed by WW2 are nothing to sneeze at either.