  In the past decade, Resonant Inelastic X-ray Scattering (RIXS) has made
remarkable progress as a spectroscopic technique. This is a direct result of
the availability of high-brilliance synchrotron X-ray radiation sources and of
advanced photon detection instrumentation. The technique's unique capability to
probe elementary excitations in complex materials by measuring their energy-,
momentum-, and polarization-dependence has brought RIXS to the forefront of
experimental photon science. We review both the experimental and theoretical
RIXS investigations of the past decade, focusing on those determining the
low-energy charge, spin, orbital and lattice excitations of solids. We present
the fundamentals of RIXS as an experimental method and then review the
theoretical state of affairs, its recent developments and discuss the different
(approximate) methods to compute the dynamical RIXS response. The last decade's
body of experimental RIXS data and its interpretation is surveyed, with an
emphasis on RIXS studies of correlated electron systems, especially transition
metal compounds. Finally, we discuss the promise that RIXS holds for the near
future, particularly in view of the advent of x-ray laser photon sources.

Strongly Correlated Electrons;Materials Science