The physics of x-ray free-electron lasers

X-ray free-electron lasers (x-ray FELs) give us for the first time the possibility to explore structures and dynamical processes of atomic and molecular systems at the angstrom-femtosecond space and time scales. They generate coherent photon pulses with time duration of a few to 100 fs, peak power of 10 to 100 GW, over a wavelength range extending from about 100 nm to less than 1 angstrom. Using these novel and unique capabilities new scientific results are being obtained in atomic and molecular sciences, in areas of physics, chemistry, and biology. This paper reviews the physical principles, the theoretical models, and the numerical codes on which x-ray FELs are based, starting from a single electron spontaneous undulator radiation to the FEL collective instability of a high density electron beam, strongly enhancing the electromagnetic radiation field intensity and its coherence properties. A short review is presented of the main experimental properties of x-ray FELs, and the results are discussed of the most recent research to improve their longitudinal coherence properties, increase the peak power, and generate multicolor spectra.