The development from a test facility for the TESLA project of a linear collider with an integrated x-ray free-electron laser (FEL) to the world's first soft x-ray FEL user facility is described. In the wavelength range from 6.5 to 60 nm, FLASH (Free-Electron Laser in Hamburg) provides short pulses (similar to 25 fs) containing similar to 10(12) photons in laterally coherent beams with brilliance of about nine orders of magnitude higher than achieved at the best synchrotron radiation storage ring facilities today. FLASH has opened up a new area in photon science where matter in non-equilibrium states is studied with atomic resolution in space and time.