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.