The design and performance of Transition Radiation Trackers (TRTs) based on Kapton straw drift tubes is described. A test beam prototype of the ATLAS TRT is discussed, together with extensive Monte Carlo (MC) simulations for the full-size detector performance in real LHC conditions. These simulations have shown that the ATLAS TRT will provide powerful pattern recognition and excellent electron identification, even at the highest LHC luminosities. In the case of the HERA-B TRT the MC simulations demonstrate a good electron identification capability even at the pre-trigger level. The application of the TRT for B-physics experiments is most important, since the electron tracking and identification capabilities reduce the background from hadrons, photon conversions and fake tracks by a factor of 10-100 beyond that from the e.m. calorimeter and tracker alone.