In a free-electron laser (FEL) the lasing medium is a high-energy beam of electrons flying with relativistic speed through a periodic magnetic field. The interaction between the synchrotron radiation that is produced and the electrons in the beam induces a periodic bunching of the electrons, greatly increasing the intensity of radiation produced at a particular wavelength. Depending only on a phase match between the electron energy and the magnetic period, the wavelength of the FEL radiation can be continuously tuned within a wide spectral range. The FEL concept can be adapted to produce radiation wavelengths from millimetres to Ångstroms, and can in principle produce hard x-ray beams with unprecedented peak brightness, exceeding that of the brightest synchrotron source by ten orders of magnitude or more. This paper focuses on short-wavelength FELs. It reviews the physics and characteristic properties of single-pass FELs, as well as current technical developments aiming for fully coherent x-ray radiation pulses with pulse durations in the 100 fs to 100 as range. First experimental results at wavelengths around 100 nm and examples of scientific applications planned on the new, emerging x-ray FEL facilities are presented.