In the years leading up to the Second World War, Nazi Germany was engaged in many areas of research with military applications. Germany was lucky enough in those days to possess some of the top scientists in the world, (though the Nazi abuse of Jews drove the really smart ones away, like Albert Einstein). However, they retained other scientists, such as Wernher von Braun (the famed rocket scientist), Walter Horten the airplane designer, Werner Heisenberg, one of the heads of Germany's atomic bomb program, as well as Johannes Stark, a Nazi Doctor that was one of the proponents of Aryan science as opposed to Jewish science.
Nazi science was often in advance of that of the United States and Great Britain during World War Two. The Germans were the first to develop long range missiles, in the form of the V-1 jet-powered 'buzz' bomb, as well as the V-2 vertical-takeoff, ethanol powered rocket. After the war, Germany's rocket program was gutted by both the United States and the Soviet Union, who each wanted to be the ones to employ such useful weapons of war. After the war, the V-2s became the basis for the space programs of both the USA and USSR.
Germany was also the first to deploy jet aircraft in significant quantities during World War Two. The Messerschmitt 262 (Me-262) jet fighter/interceptor, as well as the Arado 234 (Ar-234) bomber/recon jet.
The Me-262 could reach near-supersonic speeds, and cruised at over 100 miles per hour faster than the fastest Allied planes. Equipped with light guns, and a revolutionary rocket rack for attacking densely packed bomber formations, the Me-262 was a force to be reckoned with. In fact, the Luftwaffe was experimenting with various techniques of guided missiles to attach to the Me 262. Though fortunately, the Me-262 did not reach production until late in the war.
The Ar-234 was a two engine light bomber, capable of striking a target with its heavy bomb payload very quickly, it was extremely effective when it was used in that role. However, the Ar-234 did not reach significant levels of production before the end of the war.
The Nazis also conducted numerous, brutal, and largely fruitless medical experiments (see my Nazi Medical Experiments node for more information).
Several different takes on the German atomic bomb program state that the Germans were either very near to producing an atomic weapon, or that they were rather far from producing a finished weapon. Doctor Heisenberg too some credit for 'delaying' the completion of the weapon, though it is thought that he just didn't know how to complete it. It is commonly thought that the first use of a German atomic bomb would have been the city of New York, in North America. The bomb was to be delivered using the Horten Flying Wing, a long range stealth bomber (stealth because of its small radar profile) that the Nazis had produced in small quantities.
But the Nazis never completed a weapon. However, before the end of the war, they sent a great deal of Uranium-235 to Japan (the Empire of) aboard a U-boat, which was fortunately intercepted somewhere in the Indian Ocean after Germany surrendered.
And then there is Nazi Pseudoscience, in which the commonly perceived 'advanced' state of Nazi Germany led to various speculations in the fifties about what ELSE the Nazis may have accomplished. This is where the myths of Nazi supermen and the like originated from.