  Relativistic interaction of short-pulse lasers with underdense plasmas has
recently led to the emergence of a novel generation of femtosecond x-ray
sources. Based on radiation from electrons accelerated in plasma, these sources
have the common properties to be compact and to deliver collimated, incoherent
and femtosecond radiation. In this article we review, within a unified
formalism, the betatron radiation of trapped and accelerated electrons in the
so-called bubble regime, the synchrotron radiation of laser-accelerated
electrons in usual meter-scale undulators, the nonlinear Thomson scattering
from relativistic electrons oscillating in an intense laser field, and the
Thomson backscattered radiation of a laser beam by laser-accelerated electrons.
The underlying physics is presented using ideal models, the relevant parameters
are defined, and analytical expressions providing the features of the sources
are given. Numerical simulations and a summary of recent experimental results
on the different mechanisms are also presented. Each section ends with the
foreseen development of each scheme. Finally, one of the most promising
applications of laser-plasma accelerators is discussed: the realization of a
compact free-electron laser in the x-ray range of the spectrum. In the
conclusion, the relevant parameters characterizing each sources are summarized.
Considering typical laser-plasma interaction parameters obtained with currently
available lasers, examples of the source features are given. The sources are
then compared to each other in order to define their field of applications.
