Airborne operation
Japan built a combat strike force of 825 gliders but never committed it to battle.

Allied operations: Ironically, the battle that ended Germany's paratrooper operations had the opposite effect on the Allies. Convinced of the effectiveness of airborne assaults after Crete, the Allies hurried to train and organize their own airborne units. The British established No.1 Parachute Training School at RAF Ringway near Manchester, which trained all 60,000 European paratroopers recruited by the Allies during World War II.
An Airlanding School was also set up in New Delhi, India, in October/November 1941, at the then-Welllingdon Airport (now the defunct Safdarjang Airport) to train paratroopers for the British Indian Army which had been authorised to raise an airborne-capable formation earlier, resulting in the formation of the 50th Indian Parachute Brigade. The Indian airborne forces expanded during the war to the point that an airborne corps was planned bringing together the 2nd Indian Airborne Division and the British 6th Airborne Division, but the war ended before it could materialize.
A fundamental decision was whether to create small airborne units to be used in specific coup-de-main type operations, or to organize entire airborne divisions for larger operations. Many of the early successful airborne operations were small, carried out by a few units, such as seizing a bridge. After seeing success of other units and observing smokejumper training methods on how training can be done in June 1940, General William C. Lee of the U.S. Army established the Army's first airborne division. The 101st would be reorganized into the 101st Airborne Division.
The Allies eventually formed two British and five American divisions: the British 1st and 6th Airborne Divisions, and the U.S. 11th, 13th, 17th, 82nd, and 101st Airborne Divisions. By 1944, the British divisions were grouped into the 1st Airborne Corps under Lieutenant-General Sir Frederick Browning, while the American divisions in the European Theatre (the 17th, 82nd, and 101st) were organized into the XVIII Airborne Corps under Major General Matthew Ridgway. Both corps fell under the First Allied Airborne Army under U.S. Lieutenant General Lewis H. Brereton.
The first U.S. airborne operation was by the 509th Parachute Infantry Battalion in November 1942, as part of Operation Torch in North Africa. The U.S. 82nd and 101st Airborne Divisions saw the most action in the European Theater, with the former in Sicily and Italy in 1943, and both in Normandy and the Netherlands in 1944. The 517th Parachute Regimental Combat Team was the principal force in Operation Dragoon in Southern France. The 17th Airborne Division deployed to England in 1944 but did not see combat until the Battle of the Bulge in January 1945 where they, along with the 82nd and 101st Airborne Divisions were deployed as ground troops.
The U.S. 11th and 13th Airborne Divisions were held in reserve in the United States until 1944 when the 11th Airborne Division was deployed to the Pacific, but mostly used as ground troops or for smaller airborne operations. The 13th Airborne Division was deployed to France in January 1945 but never saw combat as a unit.

Soviet operations: The Soviets mounted only one large-scale airborne operation in World War II, despite their early leadership in the field in the 1930s. Russia also pioneered the development of combat gliders, but used them only for cargo during the war.
Axis air superiority early in the conflict limited the ability of the Soviets to mount such operations, whilst later in the conflict ongoing shortages of materiel, including silk for parachutes, was also a problem. Nonetheless, the Soviets maintained their doctrinal belief in the effectiveness of airborne forces, as part of their concept of "deep battle", throughout the war. The largest drop during the war was corp-sized (the Vyazma airborne Operation, the 4th Airborne Corps). It was unsuccessful. Airborne formations were used as elite infantry units however, and played a critical role in several battles. For example, at the Battle of Kursk, the Guards Airborne defended the eastern shoulder of the southern penetration and was critical to holding back the German penetration.
The Soviets sent at least one team of observers to the British and American airborne planning for D-Day, but did not reciprocate the liaison.

Early commando raids: 

Operation Colossus: Raid on the Tragino Aqueduct: Britain's first airborne assault took place on February 10, 1941, when 'X' Troop, No 11 Special Air Service Battalion (which was formed from No 2 Commando and subsequently became 1st Battalion, The Parachute Regiment) dropped into southern Italy from converted Whitley bombers flying from Malta and demolished a span of the aqueduct near Tragino in a daring night raid named Operation Colossus.

Operation Squatter: Raid on Axis airfields in Libya: 54 effectives of 'L' Detachment, Special Air Service Brigade (largely drawn from the disbanded Layforce) mounted a night parachute insertion onto two drop zones in Bir Temrad, North Africa on the night of November 16/17 1941 in preparation for a stealthy attack on the forward airfields of Gambut and Tmimi in order to destroy the Axis fighter force on the ground before the start of Operation Crusader, a major offensive by the British Eighth Army.

Operation Biting: The Bruneval raid: A Würzburg radar site on the coast of France was attacked by a company of 120 British paratroopers from 2 Battalion, Parachute Regiment, commanded by Major John Frost, in Operation Biting on February 27, 1942. The key electronic components of the system were dismantled by an English radar mechanic and brought back to Britain for examination so that countermeasures could be devised. The result was a British victory. Of the 120 paratroopers who dropped in the dead of night, there were two killed, six wounded, and six captured.

Mediterranean: 

Operation Mercury: Crete: This was the last large-scale airborne assault by Hitler and the Germans. The German paratroopers had such a high casualty rate that Hitler forbade any further large-scale airborne attacks. The Allies, on the other hand, were very impressed by the potential of paratroopers, and started to build their own airborne divisions.

Operation Torch: North Africa: The first United States airborne combat mission occurred during Operation Torch in North Africa on 8 November 1942. 531 men of the 2nd Battalion, 509th Parachute Infantry Regiment flew over 1,600 miles (2,600 km) at night from Britain, over Spain, intending to drop near Oran and capture two airfields. Navigation errors, communications problems, and bad weather scattered the forces. Seven of the 39 C-47s landed far from Oran from Gibraltar to Tunisia, and only ten actually delivered their troops by parachute drop. The remainder off-loaded after 28 C-47 troop carriers, short on fuel, landed on the Sebkra d'Oran dry lake, and marched overland to their objectives.
One week later, after repacking their own chutes, 304 men of the battalion conducted a second combat jump on 15 November 1942 to secure the airfield at Youk-les-Bains near the Tunisian border. From this base, the battalion conducted combined operations with various French forces against the German Afrika Korps in Tunisia. A unit of French Algerian infantry, the 3rd Regiment of Zouaves, was present at Youk-les-Bains and awarded the American paratroopers their own regimental crest as a gesture of respect. This badge was awarded to the battalion commander on 15 November 1942 by the 3rd Zouaves' regimental commander, and is worn today by all members of the 509th Infantry.

Operation Husky: Sicily: As part of Operation Husky, the Allied invasion of the island of Sicily, four airborne operations (two British and two American) were carried out, landing during the nights of July 9 and 10 1943. The American paratroopers were from the 82nd Airborne Division, mainly Colonel James Gavin's 505th Parachute Regimental Combat Team (consisting of the 3rd Battalion of the 504th PIR, Company 'B' of the 307th Airborne Engineer Battalion and the 456th Parachute Field Artillery Battalion, with other supporting units), making their first combat jump. Strong winds encountered en route blew the dropping aircraft off course and scattered them widely. The result was that around half the paratroopers failed to make it to their rallying points. The British airborne troops from the 1st Airborne Division were glider infantry of the 1st Airlanding Brigade, commanded by Brigadier Philip Hicks, and they fared little better. Only 12 out of 137 gliders in Operation Ladbroke landed on target, with more than half landing in the sea. Nevertheless, the scattered airborne troops maximised their opportunities, attacking patrols and creating confusion wherever possible.