  Weakly magnetized, relativistic collisionless shock waves are not only the
natural offsprings of relativistic jets in high-energy astrophysical sources,
they are also associated with some of the most outstanding displays of energy
dissipation through particle acceleration and radiation. Perhaps their most
peculiar and exciting feature is that the magnetized turbulence that sustains
the acceleration process, and (possibly) the secondary radiation itself, is
self-excited by the accelerated particles themselves, so that the phenomenology
of these shock waves hinges strongly on the microphysics of the shock. In this
review, we draw a status report of this microphysics, benchmarking analytical
arguments with particle-in-cell simulations, and extract consequences of direct
interest to the phenomenology, regarding in particular the so-called
microphysical parameters used in phenomenological studies.
