The Germans in WWII were at the forefront of industrialized warfare. They produced the first jet-powered bomber, developed the first tilt-rotor plane, and discovered fission. In most cases, Allied scientists and planners struggled to, through long hours of research and experimentation, close the technological gaps exposed by German advances. When possible though, they just stole everything they could find and called it a day.
1. Airborne Operations
The first airborne operations in combat were all executed by Germans during invasions of European countries. Normandy, Denmark, France, and the Netherlands all fell quickly while small units of German paratroopers seized key infrastructure or destroyed enemy defenses ahead of the main army. In the Battle of Crete, British intelligence operatives were able to determine the exact locations that German paratroopers would land and inflicted heavy losses. Adolf Hitler banned future large-scale airborne operations, but Britain and America were impressed by the ability of the airborne to complete their mission despite the losses. The Allies drastically stepped up their training and organizing of airborne units. The paratroopers they trained contributed decisively to the successful invasions of Sicily and Normandy.
Synchropter is a specific class of helicopter, one that uses intermeshing blades that turn in opposite directions. An unmanned version is being evaluated for medical evacuation missions by the Marine Corps. The HH-43 was a Navy, Marine Corps, and Air Force synchropter used from the 1950s-1970s as a rescue and firefighting helicopter. Designs for both helicopters borrow heavily from a Fleittner Fl 282 recovered during Operation Lusty. Allied aviators didn't just benefit from recovering the helicopter though. They also got the designer, Anton Flettner through Operation Paperclip.
3. Jet-powered aircraft
The Messerschmitt Me 262 was the first jet airplane used in combat and it was very effective against Allied bomber formations. Both the U.S. and the Soviet Union seized Me 262s as they captured German territory and reverse engineered the German planes. While neither country would finish building jet aircraft during the war, when American F-86 Sabres later faced off against Soviet MiG-15s in MiG Alley over Korea, it was a fight between Me 262 descendants. Similarly, the U.S. captured the Arado Ar 234 jet-powered bomber. Technology from the Arado would go on to be found in the U.S. Army Air Force's B-45s and B-47s.
4. Cruise missiles
In June 1944, V-1 flying bombs started raining down on London. The V-1, "the buzz bomb", was inaccurate but took a psychological toll on the British. The U.S. wanted its own version in preparation for the invasion of mainland Japan, and so recovered pieces of crashed and detonated V-1s. By September, it had successfully tested the JB-2 Loon, a virtual copy of the V-1. The JB-2 was never fired in combat since nuclear weapons were dropped first and Japan surrendered. Technology from the V-1 would later appear in the MGM-1 Matador, though the Matador would use a turbojet instead of the pulse jet that gave the V-1 its signature buzzing sound.
Meth was invented in 1893 by a Japanese chemist, but it was first used in war by WWII Germany. News of a wonder drug that kept tankers and pilots awake crossed to the Allies who wanted to find a way to save their crews as well. Tests on the Allied side went badly though, and the Allies stopped giving the drug to pilots. Ground soldiers still used it to overcome fatigue.
Rocket science was one of the key areas of interest during Operation Paperclip. Famously, the scientists who pioneered the U.S. and Soviet space programs were taken from Germany in the final months and years immediately after the war. At first, both the U.S. and Soviets constructed their own V-2 bombs before kicking off the space race in earnest. The stolen V-2s and their creators paved the way for U.S. rocket programs from the Redstone rockets to the Saturn and Apollo missions. The Saturn rocket, used in the Apollo program, is the only rocket that has carried a man outside of low earth orbit.