Relativistic outflows in high-energy astrophysical sources are copious emitters of non-thermal radiation. These outflows are most likely magnetically dominated, i.e. the magnetic energy per particle greatly exceeds the rest mass. Since the spatial scale of the flow and the dissipative scales are separated by several orders of magnitude (i.e. the Reynolds numbers are huge), turbulence is a natural candidate to dissipate the magnetic energy and accelerate a population of non-thermal particles. The accelerated particles could emit the observed radiation via synchrotron and inverse Compton cooling. The advent of large scale Particle-In-Cell simulations makes it possible to study the radiative turbulent energy cascade from first physical principles. I will show that particles are energised in large scale current sheets where the magnetic field reconnects. The energised particles have a strong pitch angle anisotropy, i.e. their velocity is nearly aligned with the local magnetic field. I will show that the pitch angle anisotropy hardens the spectrum of the synchrotron radiation, and suppresses the synchrotron luminosity with respect to the inverse Compton luminosity. I will discuss the implications for the modelling of non-thermal radiation from blazars and Pulsar Wind Nebulae.