  The mechanism of diffusive Fermi acceleration at collisionless plasma shock
waves is widely invoked in astrophysics to explain the appearance of
non-thermal particle populations in a variety of environments, including sites
of cosmic ray production, and is observed to operate at several sites in the
heliosphere. This review outlines the principal results from the theory of
diffusive shock acceleration, focusing first on how it produces power-law
distributions in test-particle regimes, where the shock dynamics are dominated
by the thermal populations that provide the seed particles for the acceleration
process. Then the importance of non-linear modifications to the shock
hydrodynamics by the accelerated particles is addressed, emphasizing how these
subsequently influence non-thermal spectral formation.
