Since the pioneering work of Tajima and Dawson on the laser electron accelerator, experiments on the excitation of high-amplitude plasma waves by beat-wave, photon wakefield and self-resonant wakefield have been performed in plasmas and give evidence of electron acceleration in the range 1 - 100 MeV for distances of less than one centimetre. Electrons, either from the plasma itself, or injected from outside, are accelerated to these relativistic energies by the longitudinal electric field of the laser-excited plasma wave whose phase velocity is close to the speed of light. After a short survey of the excitation mechanism of this relativistic plasma wave, the scaling laws which link the electric field of the plasma wave, the energy gain of accelerated electrons and the acceleration length to the parameters of the laser will be recalled. Then, limits for the energy of accelerated electrons will be given in the case of passing and trapped electrons as well as in the case of wavebreaking. These classical results will be compared with recent experimental results of electron acceleration in CO2 and Nd, beat-wave, as well as for self-resonant wakefield. Future experiments on photon wakefield will be discussed. Relevant work performed to improve the energy gain of acceleration schemes such as light channelling by preformed plasmas or by relativistic effects will be discussed.