The Me 262 Project are a group of Americans dedicated to reproducing the first jet fighter to ever see combat. They're building two different versions of the Messerschmitt 262, a two seat trainer and a single seat fighter. One of the single seat aircraft has already been shipped to Germany and will be flown in the next ILA-Berlin airshow. It is the first 262 to fly over germany in 60 years.
Original 262s still exist although there are only a handful of them, but none of them are flying. The original aircraft was never very reliable and the loss of a single original would be too much of a loss. So reproductions are really the way to go here.
Their site tells the whole build story of the new aircraft. One of the interesting things is that they have been willing to make some design changes to increase reliability for the airshow circuit. The landing gear has been subtilely improved. More importantly, the plane's engines are modern General Electric J-85s instead of the original BMW and Jumo units. The original engines were notoriously problematic unlike the modern J-85. The shape of the engine nacelle exteriors will remain, including the original "bullet" pre-compressor but modern engines will be placed inside of them.