The USSR also tried to expand its sphere of influence. To begin with, in 1950s, it had to face the resistance of countries like Germany, Hungary and Czechoslovakia. It sent its armies to dismiss unfriendly governments and install friendly governments in those countries. After 1960s, China decided to disassociate itself from the USSR and the latter tried unsuccessfully to pressurise China. In 1971, USSR invaded Afghanistan to install a friendly government there. The US in turn provided armed support to Afghan rebels who were also religious extremists. A prolonged civil war ensued and as the USSR decided to withdraw from Afghanistan in 1985, the country fell to religious extremists led by the Taliban which has now turned against the USA. In all this chaos, we can see that the threat of warfare and destruction was faced by the third world which was trying to emerge out of colonial control and the wars were not fought so much by the Soviet or US troops but by the people of the third world.
Both USA and USSR were in possession of nuclear weapons but knew very well that neither would be a winner in a nuclear war. Yet, they formed military and strategic alliances - the west formalised its alliances in an organisation known as North Atlantic Treaty Organisation (NATO) in 1949. To counter this, Communist nations made similar alliances and signed the Warsaw Pact. In addition to this, U.S. established regional military and strategic alliances like South East Asian Treaty Organisation (SEATO) and Central Treaty Organisation (CENTO).
Even though they strengthened themselves by creating military blocks, they knew that fighting wars would lead to a massive destruction of human life, threatening the whole civilisations. Hence, they maintained a constrained relationship acting as a deterrent to each other.
The alliances helped the super powers which wanted to expand their influence to gain access to :