From Wikipedia
Cold War
Episode Eight
The nuclear arms race was a competition for supremacy in nuclear warfare between the United States, the Soviet Union, and their respective allies during the Cold War. During the Cold War, in addition to the American and Soviet nuclear stockpiles, other countries also developed nuclear weapons, though none engaged in warhead production on nearly the same scale as the two superpowers.

U.S. and USSR/Russia nuclear stockpiles.
In the years immediately after the Second World War, the United States had a nuclear monopoly on both specific knowledge and, most importantly, raw materials. Initially, it was thought that uranium was relatively rare in the world, but this was discovered to be incorrect. While American leaders hoped that their exclusivity would be able to draw concessions from the Soviet Union, this proved ineffective. Behind the scenes the Soviet regime was working furiously to build their own atomic weapons. During the war Soviet efforts had been limited by a lack of uranium, but new supplies in Eastern Europe were taken and provided a steady supply while the Soviets developed a domestic source. While American thinkers had predicted that the USSR would not have nuclear weapons until the mid-1950s, the first Soviet bomb was detonated on August 29 of 1949, shocking the entire world. The weapon (called "Joe One" by the West) was more or less a copy of the weapon which the United States had dropped on Japan ("Fat Man").
Governments devoted massive amounts of resources to increasing the quality and quantity of their nuclear arsenal. Both nations quickly began work on hydrogen bombs and the United States detonated the first such device on November 1, 1952. Again the Soviets surprised the Americans by exploding a deployable thermonuclear device of their own the next August, though it was not actually a "true" multi-stage hydrogen bomb (that would wait until 1955). The Soviet H-bomb was almost completely a product of domestic research, as their espionage sources in the USA had only worked on very preliminary (and incorrect) versions of the hydrogen bomb.

A chart of the Space Race as driven by the nuclear threat, graphing how the U.S. started behind but eventually caught up and surpassed the Soviet Union.
The most important development in terms of delivery in the 1950s was the introduction of ICBMs. Missiles had long been seen as the ideal platform for nuclear weapons and in 1957 on the 4th of October with the launch of Sputnik the Soviet Union showed the world that they had missiles that could hit anywhere in the world. The United States launched their own on the 31 October 1959. The Space Race showcased technology that was critical to nuclear weapons delivery (the ICBM boosters) while maintaining the Beginning and ages of science and exploration.
This period also saw attempts begin to defend against nuclear weapons. Both powers built large radar arrays to detect incoming bombers and missiles. Fighters to use against bombers and anti-ballistic missiles to use against ICBMs were also developed. Large underground bunkers were constructed to save the leadership of the superpowers, and individuals were told to build fallout shelters and taught how to react to a nuclear attack (civil defense). These bombs could kill millions in the event of an attack by either side.

The Titan II Intercontinental ballistic missile (ICBM) carried a 9 Mt W53 warhead, one of the most powerful nuclear weapons fielded by the United States during the Cold War.