Particle acceleration and radiation in Pulsar Wind Nebulae

Pulsar Wind Nebulae are the astrophysical sources that host the most relativistic shocks in Nature and the only Galactic sources in which we have direct evidence of PeV particles. These facts make them very interesting from the point of view of particle acceleration physics, and their proximity and brightness make them a place where fundamental processes common to different classes of relativistic sources have a better chance to be understood. I will discuss how well we understand the physics of Pulsar Wind Nebulae, describing recent progress and highlighting the main open questions. I will be mostly concerned with the subject of particle acceleration, but, as we will see, in order to clarify the physics of this process, it is important to determine the conditions of the plasma in the nebula. These in turn can only be constrained through detailed modelling of the PWN dynamics and radiation. The shock in the Crab Nebula is probably the most efficient accelerator known, both in terms of conversion of the flow energy into accelerated particles (tens of percent) and in terms of maximum energy achieved ($\approx 10^{15}$ eV). I will review the different mechanisms proposed to explain particle acceleration and recent constraints derived from the comparison of synthetic emission maps with multi-wavelength data, including variability.