Nuclear warfare
A few backyard fallout shelters were built by private individuals.
Henry Kissinger's view on tactical nuclear war in his controversial 1957 book Nuclear Weapons and Foreign Policy was that any nuclear weapon exploded in air burst mode that was below 500 kilotons in yield and thus averting serious fallout, may be more decisive and less costly in human lives than a protracted conventional war.
A list of targets made by the United States was released sometime during December 2015 by the U.S. National Archives and Records Administration. The language used to describe targets is "designated ground zeros". The list was released after a request was made during 2006 by William Burr who belongs to a research group at George Washington University, and belongs to a previously top-secret 800-page document. The list is entitled "Atomic Weapons Requirements Study for 1959" and was produced by U.S. Strategic Air Command during the year 1956.

1960s: In 1960, the United States developed its first Single Integrated Operational Plan, a range of targeting options, and described launch procedures and target sets against which nuclear weapons would be launched, variants of which were in use from 1961 to 2003. That year also saw the start of the Missile Defense Alarm System, an American system of 12 early-warning satellites that provided limited notice of Soviet intercontinental ballistic missile launches between 1960 and 1966. The Ballistic Missile Early Warning System was completed in 1964.
The most powerful atomic bomb ever made, the Tsar Bomba, was tested by the Soviets on October 30, 1961. It was 50 megatons, or equal to 50 million tons of regular explosives. A complex and worrisome situation developed in 1962, in what is called the Cuban Missile Crisis. The Soviet Union placed medium-range ballistic missiles 90 miles (140 km) from the United States, possibly as a direct response to American Jupiter missiles placed in Turkey. After intense negotiations, the Soviets ended up removing the missiles from Cuba and decided to institute a massive weapons-building program of their own. In exchange, the United States dismantled its launch sites in Turkey, although this was done secretly and not publicly revealed for over two decades. First Secretary Nikita Khrushchev did not even reveal this part of the agreement when he came under fire by political opponents for mishandling the crisis. Communication delays during the crisis led to the establishment of the Moscow–Washington hotline to allow reliable, direct communications between the two nuclear powers.
By the late 1960s, the number of ICBMs and warheads was so high on both sides that it was believed that both the United States and the Soviet Union were capable of completely destroying the infrastructure and a large proportion of the population of the other country. Thus, by some western game theorists, a balance of power system known as mutually assured destruction (or MAD) came into being. It was thought that no full-scale exchange between the powers would result in an outright winner, with at best one side emerging the pyrrhic victor. Thus both sides were deterred from risking the initiation of a direct confrontation, instead being forced to engage in lower-intensity proxy wars.
During this decade the People's Republic of China began to build subterranean infrastructure such as the Underground Project 131 following the Sino-Soviet split.
One drawback of the MAD doctrine was the possibility of a nuclear war occurring without either side intentionally striking first. Early Warning Systems (EWS) were notoriously error-prone. For example, on 78 occasions in 1979 alone, a "missile display conference" was called to evaluate detections that were "potentially threatening to the North American continent". Some of these were trivial errors and were spotted quickly, but several went to more serious levels. On September 26, 1983, Stanislav Petrov received convincing indications of an American first strike launch against the Soviet Union, but positively identified the warning as a false alarm. Though it is unclear what role Petrov's actions played in preventing a nuclear war during this incident, he has been honored by the United Nations for his actions.
Similar incidents happened many times in the United States, due to failed computer chips, misidentifications of large flights of geese, test programs, and bureaucratic failures to notify early warning military personnel of legitimate launches of test or weather missiles. For many years, the U.S. Air Force's strategic bombers were kept airborne on a daily rotating basis "around the clock" (see Operation Chrome Dome), until the number and severity of accidents, the 1968 Thule Air Base B-52 crash in particular, persuaded policymakers it was not worthwhile.

1970s: Israel responded to the Arab Yom Kippur War attack on 6 October 1973 by assembling 13 nuclear weapons in a tunnel under the Negev desert when Syrian tanks were sweeping in across the Golan Heights. On 8 October 1973, Israeli Prime Minister Golda Meir authorized Defense Minister Moshe Dayan to activate the 13 Israeli nuclear warheads and distribute them to Israeli air force units, with the intent that they be used if Israel began to be overrun.
On 24 October 1973, as US President Richard Nixon was preoccupied with the Watergate scandal, Henry Kissinger ordered a DEFCON-3 alert preparing American B-52 nuclear bombers for war. Intelligence reports indicated that the USSR was preparing to defend Egypt in its Yom Kippur War with Israel. It had become apparent that if Israel had dropped nuclear weapons on Egypt or Syria, as it prepared to do, then the USSR would have retaliated against Israel, with the US then committed to providing Israeli assistance, possibly escalating to a general nuclear war.
By the late 1970s, people in both the United States and the Soviet Union, along with the rest of the world, had been living with the concept of mutual assured destruction (MAD) for about a decade, and it became deeply ingrained into the psyche and popular culture of those countries. 
On May 18, 1974, India conducted its first nuclear test in the Pokhran test range. The name of the operation was Smiling Buddha, and India termed the test as a "peaceful nuclear explosion."
The Soviet Duga early warning over-the-horizon radar system was made operational in 1976. The extremely powerful radio transmissions needed for such a system led to much disruption of civilian shortwave broadcasts, earning it the nickname "Russian Woodpecker".
The idea that any nuclear conflict would eventually escalate was a challenge for military strategists. This challenge was particularly severe for the United States and its NATO allies. It was believed (until the 1970s) that a Soviet tank offensive into Western Europe would quickly overwhelm NATO conventional forces, leading to the necessity of the West escalating to the use of tactical nuclear weapons, one of which was the W-70.
This strategy had one major (and possibly critical) flaw, which was soon realized by military analysts but highly underplayed by the U.S. military: conventional NATO forces in the European theatre of war were far outnumbered by similar Soviet and Warsaw Pact forces, and it was assumed that in case of a major Soviet attack (commonly envisioned as the "Red tanks rolling towards the North Sea" scenario) that NATO—in the face of quick conventional defeat—would soon have no other choice but to resort to tactical nuclear strikes against these forces. Most analysts agreed that once the first nuclear exchange had occurred, escalation to global nuclear war would likely become inevitable. The Warsaw Pact's vision of an atomic war between NATO and Warsaw Pact forces was simulated in the top-secret exercise Seven Days to the River Rhine in 1979. The British government exercised their vision of a Soviet nuclear attack with Square Leg in early 1980.
Large hardened nuclear weapon storage areas were built across European countries in anticipation of local US and European forces falling back as the conventional NATO defense from the Soviet Union, named REFORGER, was believed to only be capable of stalling the Soviets for a short time.

1980s: In the late 1970s and, particularly, during the early 1980s under U.S. President Ronald Reagan, the United States renewed its commitment to a more powerful military, which required a large increase in spending on U.S. military programs. These programs, which were originally part of the defense budget of U.S. President Jimmy Carter, included spending on conventional and nuclear weapons systems. Under Reagan, defensive systems like the Strategic Defense Initiative were emphasized as well.
Another major shift in nuclear doctrine was the development and the improvement of the submarine-launched, nuclear-armed, ballistic missile, or SLBM. It was hailed by many military theorists as a weapon that would make nuclear war less likely. SLBMs—which can move with "stealth" (greatly lessened detectability) virtually anywhere in the world—give a nation a "second strike" capability (i.e., after absorbing a "first strike"). Before the advent of the SLBM, thinkers feared that a nation might be tempted to initiate a first strike if it felt confident that such a strike would incapacitate the nuclear arsenal of its enemy, making retaliation impossible. With the advent of SLBMs, no nation could be certain that a first strike would incapacitate its enemy's entire nuclear arsenal. To the contrary, it would have to fear a near-certain retaliatory second strike from SLBMs. Thus, a first strike was a much less feasible (or desirable) option, and a deliberately initiated nuclear war was thought to be less likely to start.