Astrophysical compact objects, neutron stars and black holes, are powerful sources of non-thermal electromagnetic emission spanning many orders of magnitude in photon energy, from radio waves to multi-TeV gamma-rays. Despite multiple groundbreaking observational discoveries done in recent years, our understanding of the dynamics of relativistic plasmas that produce these emission signatures remains limited. In this talk I will describe a few successful examples of modeling the observed light coming from these remarkable astrophysical laboratories using first-principles numerical simulations.