Air supremacy
Levels: Air supremacy is the highest level, where a side holds complete control of the skies. It is defined by NATO and the United States Department of Defense as the "degree of air superiority wherein the opposing air force is incapable of effective interference".
Air superiority is the second level, where a side is in a more favorable position than the opponent. It is defined in the NATO glossary as the "degree of dominance in [an] air battle ... that permits the conduct of operations by [one side] and its related land, sea and air forces at a given time and place without prohibitive interference by opposing air forces."
Favorable air situation is defined as "an air situation in which the extent of air effort applied by the enemy air forces is insufficient to prejudice the success of friendly land, sea or air operations."
Air parity is the lowest level of control, where no side holds any level of control of skies.

Methods: Although the destruction of enemy aircraft in air-to-air combat is the most obvious aspect of air superiority, it is not the only method of obtaining air superiority. Historically, the most effective method of gaining air superiority is the destruction of enemy aircraft on the ground and the destruction of the means and infrastructure by which an opponent may mount air operations (such as destroying fuel supplies, cratering runways with anti-runway penetration bombs and the sowing of air-fields with area denial weapons). A historical example of this is Operation Focus in which the outnumbered Israeli Air Force dealt a crippling blow to the Egyptian, Jordanian and Syrian Air Forces and airfields at the start of the Six-Day War, achieving Israeli air supremacy.
Disruption can be carried out through ground and air attack. The main role for which the British Special Air Service was formed was to conduct raids on German aircraft and airfields. During operations in the Western Desert the SAS are reckoned to have destroyed more than 400 enemy aircraft. On 6 December 1944, the Imperial Japanese Army Air Force Raiding Group Teishin Shudan destroyed B-29 aircraft on Leyte. During the Cold War, the Soviet Union claimed it could achieve air superiority despite the inferiority of its fighters, by over-running NATO airfields and parking their tanks on the runways, similar to what they have done during Tatsinskaya Raid during the Battle of Stalingrad (note the Germans used parts of their autobahn motorways as airfields during the last war). The Soviet Union planned to use its Spetsnaz special forces in attacks on NATO airfields in the event of conflict.
Attacks by special forces have been seen by some commanders as a way to level the playing field when faced by superior numbers or technology. Given the disparity in effectiveness between their own and South Korean and US fighters, North Korea maintains a large force of infiltration troops; in the event of a war, they would be tasked, among other missions, with attacking coalition airfields with mortar, machine gun and sniper fire, possibly after insertion by some 300 An-2 low radar-observable biplanes. This strategy has been practiced in active conflicts even in recent decades; during the asymmetrical warfare of the War in Afghanistan, 15 fedayeen destroyed or severely damaged eight United States Marine Corps Harrier jump jets in the September 2012 Camp Bastion raid, one result of which being pilots fighting as infantry for the first time in 70 years. Similarly, during the Iraqi War, four Apaches were destroyed on the ground in 2007 by insurgents armed with mortar, which were unintentionally aided by web-published geotagged photographs taken by coalition soldiers.

History: 

First World War: The First World War saw many firsts in the field of aerial warfare, including the deployment of aircraft armed with machine guns, the first successful engagement involving synchronisation-gun-armed aircraft on the afternoon of 1 July 1915. Throughout the conflict, air superiority on the Western Front changed hands between the German Empire and the Allies several times. It became recognised that the worst losses was amongst new pilots, many of whom lasted just a day or two. The emergence of specialised fighter units, which were typically led by highly experienced pilots, some of them survivors of the Fokker Scourge period, greatly increased the effectiveness of fighter units.
Early on, the Allies gained a lead over the Germans by introducing machine-gun armed types such as the Vickers F.B.5 Gunbus fighter and the Morane-Saulnier L. In response, Germany bolstered its own aerial development efforts; a major achievement of the era was the Stangensteuerung (push rod controller), a genuine synchronisation gear, developed by the Fokker company. The device was fitted to the most suitable Fokker type, the Fokker M.5K (military designation Fokker A.III), of which A.16/15, assigned to Otto Parschau, became the prototype of the Fokker Eindecker series of fighter designs. This subsequently contributed to a period of German air superiority known as the Fokker Scourge, lasting between late 1915 and early 1916. A briefer period of German aerial dominance occurred in the Bloody April of April 1917; paradoxically, the Germans were disadvantaged on paper during Bloody April in terms of numerical inferiority; their effectiveness was increased by confining themselves to mainly operating over friendly territory, both reducing the possibility of pilots being captured and increasing the amount of time they could stay in the air. Moreover, German pilots could choose when and how they would engage, effectively dictating the terms of combat.
The Italian Corpo Aeronautico Militare established air superiority over the Austro-Hungarian Imperial and Royal Aviation Troops at the Battle of Vittorio Veneto in late October 1918. The Allies operated roughly 600 aircraft (93 Anglo-French, including four RAF squadrons) to gain complete air superiority in this final offensive. The defeat suffered by Austria-Hungary at Vittorio Veneto has been attributed with causing the dissolution of the empire. In turn, the surrender of Germany's primary ally was another major factor in the German Empire's decision that the conflict was no longer viable and needed to end.

Interwar period: In 1921, Italian aerial warfare theorist Giulio Douhet published The Command of the Air, a book positing that future wars would be decided in the skies. At the time, mainstream military theory did not see air power as a war-winning tactic. Douhet's idea was that air power could be a decisive force and be used to avoid the long and costly War of Attrition. In The War of 19, Douhet theorized that a future war between Germany and France would be settled in a matter of days, as the winner would be the one to gain air supremacy and destroy a few enemy cities with aerial bombs. He speculated that, while the targets would be announced ahead of time and all the population evacuated, but that the event would terrorize citizens into pressuring their government into immediate surrender. At the beginning of the Second World War, Douhet's ideas were dismissed by some, but it became apparent that his theories on the importance of aircraft were supported by events as the war continued.
In 1925, the Royal Air Force (RAF) tested the ability of air supremacy in isolation from other warfare forms during their first independent action in Waziristan. The operation, that later came to be known as Pink's War after Wing Commander Richard Pink in charge, used only air warfare in a combination of air attack and air blockade over 54 days to force militant tribes to surrender. The campaign was successful in defeating the tribes with two deaths for the RAF, but contemporary critics were not entirely convinced of its use in isolation; Commander-in-Chief, India General Sir Claud Jacob stated that "satisfactory ... the results of these operations have been, I am of [the] opinion that a combination of land and air action would have brought about the desired result in a shorter space of time, and next time action has to be taken, I trust that it will be possible to employ the two forces in combination".

American general Billy Mitchell was another influential air power theorist of the inter-war period. After the First World War I, then-Assistant Chief of Air Service in the United States Army Air Service under Chief Mason Patrick, Mitchell arranged live fire exercises that proved that aircraft could sink battleships (the largest and most heavily armed class of warships). The first of these was Project B in 1921, in which the captured First World War-era German battleship, SMS Ostfriesland, was sunk by a flight of bombers in 22 minutes.
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.