Most of the visible matter in the Universe is a plasma, that is a dilute gas of ions, electrons, and neutral atoms. In many circumstances, the dynamics of this plasma can be modeled in the continuum limit, using the equations of fluid mechanics. Unlike most terrestrial flows, however, astrophysical plasmas are highly compressible and strongly affected by additional physics such as magnetic fields, radiation transport, and gravity. I will introduce the kind of problems astrophysicists study using fluid mechanics, discuss the importance of additional physics to model such problems realistically, and describe numerical methods that are essential for investigation of the nonlinear regime. Finally I will highlight one application of these methods: accretion of plasma into black holes.