  Kinetic approaches provide an effective description of the process of
particle acceleration at shock fronts and allow to take into account the
dynamical reaction of the accelerated particles as well as the amplification of
the turbulent magnetic field as due to streaming instability. The latter does
in turn affect the maximum achievable momentum and thereby the acceleration
process itself, in a chain of causality which is typical of non-linear systems.
Here we provide a technical description of two of these kinetic approaches and
show that they basically lead to the same conclusions. In particular we discuss
the effects of shock modification on the spectral shape of the accelerated
particles, on the maximum momentum, on the thermodynamic properties of the
background fluid and on the escaping and advected fluxes of accelerated
particles.
