Most of the observable matter in the Universe is in the form of plasma, or tenuous ionized gas, and the complicated behavior of plasmas underlies many processes in astrophysics. Plasmas interact with electromagnetic fields, display collective effects and instabilities, and can, under certain conditions, accelerate particles far out of thermal equilibrium. Plasmas are also responsible for dissipation of energy, transport, and turbulent properties of many astrophysical flows. While intrinsically microscopic, plasma processes often couple small and large spatial scales and have a way of introducing nonlinear feedback mechanisms into astrophysical systems. This behavior makes it challenging to understand and simulate such plasmas, often requiring kinetic and multiscale methods. I will discuss the state of numerical studies of plasma astrophysical processes, such as shock acceleration and reconnection, and their applications to astrophysical phenomena in gamma-ray bursts, supernova remnants, and neutron star magnetospheres.