Airborne ops are pretty much passé, although airborne troops on light equipment scales are very useful for air delivery before heavier equipped troops arrive. Airborne troops are well trained and highly motivated and can be deployed before anyone else to secure key objectives, like airfields and other facilities, for follow on forces.
The German airborne assault on Crete, while successful, was a pyrrhic victory. Losses were so high that the Germans abandoned airborne operations for the remainder of WW2, altho they did maintain paratroop formations which fought as conventional infantry. It was hard to weave airborne operations into what was essentially a series of strategic withdrawals by the Germans after 1942.
The allies took some bloody noses with their early airborne operations in N. Africa, Sicily and Italy. Problems were poor ID of DZs, dispersed drops, poor navigation, shortage of transports and losses to friendly anti-aircraft fire when transports and gliders over flew their own navies and it became a weapons free environment.
Dispersion of forces and poor ID of DZs/navigation was a big problem with the night drops into Normandy, but airborne troops did create a lot of consternation in rear areas and were useful in disrupting German command, control and comms and did block some German response to the beach landings. After that they were put back in the bottle as SHAEF strategic reserve until employment in Market-Garden.
The problems of Market-Garden are well known and include hasty planning, dispersed drops, DZs too far from the objectives, multiple lifts due to lack of aircraft, excessive time and space which precluded a quick link up with heavier ground forces, poor intelligence which ignored or minimized the presence of German armor near the Brit objectives, poor comms and overall command and control after troops were dropped, and an overly optimistic ground plan which tried to push a full corps up a single route involving crossings over several major river obstacles.
Op Varsity, the airborne component of the Brit crossing the Rhine, was the next up. It was quite cautious given the painful lessons of Market-Garden. DZs were shallow and within range of friendly artillery across the Rhine and the airborne drops only went in AFTER ground forces had crossed the Rhine. Link-up with ground troops was within 4-5 hours of the airborne drops, and it could be argued that the airborne drops weren't much of a factor in the successful Rhine crossings. On the airborne side, DZ identification and navigation were better, troops were more concentrated on DZs, and everything came in a single lift, incl re-supply.
We really haven't seen anything major since, except for a bit in Korea, and of course, the defeat of the French airborne operation at Dien bien Phu. Many armies retain some limited airborne capability in order to keep a small number of troops at peak training and readiness. The Russians have the largest numbers. The maintenance of large airborne forces is very extravagant in terms of resources and airlift. Too some extent the emphasis on light, highly mobile forces has shifted to air assault with helicopters, but these formations are also hugely expensive and present significant deployment times and large air/sea bills for deployment.
The basic limitations of airborne forces; light equipment, short staying power, limited use against a heavy mech-armor enemy and expensive bills for training, equipment and airlift still haven't gone away. But there still is a need for small numbers of highly trained, motivated and rapidly deployable light troops which the airborne can provide.