Air supremacy
Mitchell's ideas were not popular, with his outspoken opposition to Army and Navy resistance resulting in a court-martial that precipitated his resignation, but he would prove prescient; his 1924 inspection tour of Hawaii and Asia culminated in a report (published in 1925 as the book Winged Defense) that predicted future war with Japan, including the attack on Pearl Harbor. He would also go on to influence air power advocates such as Russian-American Alexander P. de Seversky, whose 1942 New York Times bestselling book, Victory Through Air Power, was made into a 1943 Walt Disney animated film that opened with a quote from Mitchell; the film is reported to have been influentially shown by Winston Churchill to Franklin D. Roosevelt in support of long-range bombing.
Seeking to influence the outcome of the Spanish Civil War, various international powers tried to influence the conflict via a heavy reliance on air power. The French government provided aircraft to the Republicans covertly, such as the Potez 540 bomber aircraft (nicknamed the "Flying Coffin" by Spanish Republican pilots), Dewoitine aircraft, and Loire 46 fighter aircraft being sent to the Republican forces, along with a group of trained fighter pilots and engineers to aid the Republicans. Also, until 8 September 1936, aircraft could freely pass from France into Spain if they were bought in other countries. The Soviet Union also covertly aided Republicans, between 634 and 806 aircraft were supplied alongside various other armaments. Both Italy and Nazi Germany supplied large numbers of aircraft to the Nationalists while also deploying their own units, such as the Condor Legion and the Aviazione Legionaria, to bolster the Nationalist's forces with their own. While the Soviet aircraft were in current service with their own forces, they proved inferior to those supplied by Germany by the end of the conflict. The use of aircraft, particularly by the Nationalists to continually pressure Republican forces and compel multiple withdrawals during the Aragon Offensive, allegedly informed both the Germans and Soviets of the value of using aircraft to support infantry.

Second World War: At the beginning of the Second World War, the opposing sides developed different views on the importance of air power. Nazi Germany viewed it as a helpful tool to support the German Army, the approach being dubbed "flying artillery". The Allies saw it, specifically long-range strategic bombing, as being a more important part of warfare which they believed capable of crippling Germany's industrial centers.
After the Battle of France, the Luftwaffe (Germany's air force) achieved air supremacy over Western Europe. The Battle of Britain represented a concerted attempt by Germany to establish air superiority over Great Britain, which it never achieved. Through home-territory advantage and Germany's failure to push home its strategy of targeting Britain's air defenses, Britain was able to establish air superiority over the territory – a superiority that it never lost. It denied the German military air superiority over the English Channel, making a seaborne invasion (planned as Operation Sea Lion) impossible in the face of Britain's naval power. Strategically, the overall situation at home and abroad at the end of the battle might be considered air parity between Britain and Germany. After the air battle, known as the Battle of Britain, the Germans switched to a strategy of night bombing raids, which Britain echoed with raids over Germany.
During Operation Barbarossa, the Luftwaffe initially achieved air supremacy over the Soviet Union. As the war dragged on, the United States joined the fight and the combined Allied air forces gained air superiority and eventually supremacy in the West. (For example, the Luftwaffe mustered 391 aircraft to oppose over 9,000 allied aircraft on D-day.) Russia did the same on the Eastern Front, meaning the Luftwaffe could not effectively interfere with Allied land operations. Achieving total air superiority allowed the Allies to carry out ever-greater strategic bombing raids on Germany's industrial and civilian centers (including the Ruhr and Dresden), and to prosecute the land war successfully on both the Eastern and Western fronts. Following the Big Week attacks in late February 1944, the new 8th Air Force commander Jimmy Doolittle permitted P-51 Mustangs to fly far ahead of the bomber formations instead of closely escorting them starting in March 1944. This commenced in March 1944 and was part of a massive "fighter sweep" tactic to clear German skies of Luftwaffe fighters. Allied planes went after the German fighters wherever they could be found and substantially lowered bomber losses for their side for the rest of the war over Western Europe.

The element of air superiority has been the driving force behind the development of aircraft carriers, which allow aircraft to operate in the absence of designated air bases. For example, the Japanese attack on Pearl Harbor was carried out by aircraft operating from carriers thousands of miles away from the nearest Japanese air base.
Some fighter aircraft specialized in combating other fighters, while interceptors were originally designed to counter bombers. Germany's most important air superiority fighters were the Messerschmitt Bf 109 and Focke-Wulf Fw 190, while the Supermarine Spitfire and Hawker Hurricane were the primary ones on the British side. Performance and range made the P-51 Mustang the outstanding escort fighter which permitted American bombers to operate over Germany during daylight hours. They shot down 5,954 aircraft, more than any other American fighter in Europe. In the Pacific Theater, the A6M Zero gave Japan air superiority for much of the early part of the war, but suffered against newer naval fighters such as the F6F Hellcat and F4U Corsair which exceeded the Zero in performance and durability. The Hellcat shot down 5,168 enemy aircraft (the second highest number), while the land-based Lockheed P-38 was third, shooting down 3,785 in all theaters.

Cold War: 

Superpower rivalry: During the Cold War, between 1946 and 1991, the US, UK, and NATO allies faced the Soviet Union, the Warsaw Pact, and its allies. Both sides engaged in an arms race of improving radar and fighter intercept capability versus the threat of intercontinental strategic bombers carrying nuclear weapons. Initially, high altitude, later combined with high supersonic speeds, was hoped to keep nuclear bombers out of range of fighters and later surface to air missiles, both of which were sometimes equipped with nuclear warheads.  In the 1960 U-2 incident an American very high altitude spy plane was shot down over the USSR with a S-75 Dvina(SA-2) long range high altitude surface to air missile largely refuting the concept of high altitude as a refuge for high-performance bomber aircraft.  US training changed to low altitude flight of bombers and unpiloted cruise missiles in the hopes of avoiding ground-based air defense radar networks by hiding in with ground clutter and terrain, thwarting attempts at air supremacy over the enemy landmass. Ballistic missiles were also introduced and were very difficult and expensive to intercept even with nuclear-armed defensive missiles.
Airborne early warning and control flying radar aircraft as well as look down shoot down radar in fighter and interceptor aircraft allowed engaging low flying invaders again tipping the balance though this was partly ameliorated by succeeding generations of electronic countermeasures.  Ultimately the US led the way in first applying stealth technology to small strike aircraft like the F-117 and stealthy nuclear cruise missiles carried in conventional bombers for standoff release before the air defenses got too thick.  The Soviet Union invested heavily in expensive to defeat intermediate and intercontinental range nuclear missiles and less on expensive to maintain patrol bombers, though they had to spend heavily on interceptors and surface to air missiles as well as radar sites to cover the huge landmass of the Soviet Union.  The US joined with Canada to organize defense of the area of Alaska, Canada, and the continental United States with North American Aerospace Defense Command or NORAD using both interceptors, some armed with the nuclear AIR-2 Genie, and a surface to air missile component, which was at one point partly nuclearized.  Development for the B-2 stealth bomber was intended for, and in anticipation of, a nuclear war and it was the first fully mature stealth aircraft to enter service. The F-22 Advanced Tactical Fighter was a stealth fighter and interceptor aircraft designed during the Cold War as a medium altitude air superiority fighter which was intended to destroy Warsaw Pact aircraft without ever being detected or engaged; both were introduced after the fall of the Soviet Union and the end of the Cold War.
Air superiority in the feared Cold War era WW-III European theater would include fighters intercepting or diverting nuclear and conventionally armed strike aircraft and ground-based air defences, some of which were developed into mobile systems which could accompany and protect armored and mechanized formations. While the Cold War never went hot directly between NATO and Warsaw Pact alliances, the US was engaged in two major limited air wars aiding allies who faced Soviet-supported enemies, with both sides using weaponry designed to fight such a conflict; the Korean and Vietnam wars.

Korean War: The Korean War represented a major turning point for aerial warfare, being the first conflict in which jet aircraft played the central role in combat. Once-formidable fighters such as the P-51 Mustang, F4U Corsair, and Hawker Sea Fury—all piston-engined, propeller-driven, and designed during the Second World War—relinquished their air superiority roles to a new generation of faster, jet-powered fighters arriving in the theater. In the initial months of fighting, the P-80 Shooting Star, F9F Panther, Gloster Meteor and other jets under the UN flag dominated the Korean People's Air Force (KPAF) propeller-driven Soviet Yakovlev Yak-9 and Lavochkin La-9s.