In the early days of WWII, the Japanese Zero fighter, which was more maneuverable than anything the Allies could field at the time, ruled the skies over the Pacific. In one battle in April of 1942, 36 Zeros attacked the British naval base at Columbo, Ceylon (modern day Sri Lanka). About 60 RAF aircraft rose to meet them, a mix of different types, many obsolete. After the battle, almost half of the RAF planes were shot down: 15 Hawker Hurricanes, 8 Fairey Swordfish, and 4 Fairey Fulmars. The Japanese lost 1 Zero.  Early in the war, the Zero enjoyed a 12-1 kill ratio. 
Allied commanders knew that their planes were no match for the Zero. So they needed to devise a way to beat a superior plane by using superior tactics. One of the first to work out a way to deal with the Zero was Claire Chennault, commander of the famous Flying Tigers. Officially called the 1st American Volunteer Group (AVG), the Flying Tigers were an assortment of pilots who volunteered to fight for China against the Japanese invasion. The best fighter they could get at the time was the Curtiss P-40 Warhawk. While it was no match for the Zero in a head-to-head fight, Chennault taught his men how to be successful with the less maneuverable Warhawk.
Edward Jablonski, in his excellent history of WWII aviation Airwar, writes:
Chennault stressed the excellent flying qualities of the Japanese pilots, although emphasizing their tendency to fly mechanically and employ a set tactical routine. Chennault also introduced the Americans to the Zero fighter, indicating its strong points as compared with the P-40: higher ceiling, superior maneuverability, and better climbing ability. But, Chennault also pointed out, the P-40 was a more rugged aircraft, could take more punishment, and, thanks to self-sealing fuel tanks, was not so readily combustible. The P-40 was also equipped with armor plate, which the Zero lacked, thanks to the Japanese pilots' insistence on high maneuverability. Also, the heavier P-40 could outlive (that is to say, outrun) the Zero. "Use your speed and diving power to make a pass, shoot and break away."
In other words, dive, shoot and run. Don't stick around to fight, because you'll likely get shot down.
The United States was drawn fully into the Pacific War after the attack on Pearl Harbor, and in the first months its pilots were hard pressed to deal with the superior Zero fighter. But then, they caught a break.
Concurrent with the Battle of Midway in early June of 1942, the Japanese occupied the islands of Attu and Kiska in the Aleutians, perhaps hoping to divert attention away from their real target, Midway Island (if it was indeed a feint, Admiral Nimitz never took the bait). On May 3, 1942, before Japanese troops moved in, a group of Japanese planes took off from the carrier Ryujo to bomb Dutch Harbor in the Aleutians. After the attack, Flight Petty Officer Tadayoshi Koga, flying an A6M2 Zero, noticed that he had been hit by ground fire and was losing fuel. He knew he would not be able to return to the Ryujo.
Flight Petty Officer Tadayoshi Koga 
Koga radioed to his flight leader that he would land at one of two islands designated for emergency landings and await rescue by a Japanese submarine. Koga chose Akutan Island. His wingmen surveyed the landing area, and they thought a safe landing could be accomplished. Koga touched down, but the aircraft immediately flipped on its back. The ground that looked solid from the air turned out to be soft, wet tundra.
Koga's Zero, as the Americans found it on Akutan
Standard procedure dictated that the other pilots should destroy the downed Zero to keep it from falling into enemy hands. But Koga's comrades thought there was a chance that he was still alive. They couldn't bring themselves to strafe their fellow pilot. What they didn't know was that Koga died instantly when the plane flipped over. The other pilots returned to the carrier, not realizing that they were leaving behind a nearly intact aircraft.
Five weeks later, the overturned Zero was spotted by a Navy reconnaissance plane and a salvage operation was undertaken. US Navy personnel made their way to the plane and found the lifeless Koga still hanging upside down in the cockpit. They buried his body, and the plane was shipped intact (the construction of the Zero prohibited the removal of the wings) to Naval Air Station, North Island, San Diego where it was repaired. Flight testing began immediately.
The very first flight exposed weaknesses of the Zero that our pilots could exploit with proper tactics. The Zero had superior maneuverability only at the lower speeds used in dogfighting, with short turning radius and excellent aileron control at very low speeds. However, immediately apparent was the fact that the ailerons froze up at speeds above two hundred knots, so that rolling maneuvers at those speeds were slow and required much force on the control stick. It rolled to the left much easier than to the right. Also, its engine cut out under negative acceleration (as when nosing into a dive) due to its float-type carburetor. 
The Akutan Zero in American markings undergoing testing at North Island 
The Navy, and later the Army, pitted the captured Zero against the best fighters of the day: the P-38 Lightning, the P-39 Airacobra, the P-51 Mustang, the F4F-4 Wildcat, the F4U Corsair. Lt. Cmdr. Eddie R. Sanders made twenty-four flights in the captured Zero on September and October of 1942.
"We now had an answer for our pilots who were unable to escape a pursuing Zero. We told them to go into a vertical power dive, using negative acceleration, if possible, to open the range quickly and gain advantageous speed while the Zero's engine was stopped. At about two hundred knots, we instructed them to roll hard right before the Zero pilot could get his sights lined up. This recommended tactic was radioed to the fleet after my first flight of Koga's plane, and soon the welcome answer came back: It works!" Sanders said, satisfaction sounding in his voice even after nearly half a century. 
It has been said that the design of the F6F Hellcat, which Grumman developed as a replacement to the F4F Wildcat, was influenced directly by the discovery of the Akutan Zero. However, the recovery of the crashed Zero came much too late in the Hellcat's development process to affect its design. The first flight of the Hellcat came in October of 1942, only one month after the Akutan Zero began flight testing in the US. By then, the Hellcat was well on its way to production, though it didn't reach the fleet until August of 1943. 
As for the plane itself, it was destroyed during a training accident in February of 1945 when an SB2C Helldiver lost control while taxiing and rammed into it, slicing the plane to bits with its propeller. Several gauges were salvaged and donated to the National Museum of the US Navy, and other small pieces now reside in the Alaska Heritage Museum and the Smithsonian National Air and Space Museum.