Compact binary X-ray sources include white dwarfs, neutron stars, and black holes that are accreting matter from a companion star. The X-ray emission from these systems is produced by the accreting matter as it flows through an accretion disk and strikes the surface of the compact object. The emitting regions have opacities dominated by electron scattering, and radiation pressure is likely to play an important role in the hydrodynamics. Strong magnetic fields greatly modify the hydrodynamics and radiation transfer in the pulsating neutron star sources. Accretion disks have complex structure, including an electron scattering corona, a cool outer region, and possibly a thick torus in their inner region. The structure and stability properties of accretion disks are only partially understood. Major problems exist with the interpretation of the spectra and luminosities of the X-ray burst sources. The pulsed X-ray emission from the pulsating binary X-ray sources probably comes from “mounds” of accreting gas at the magnetic poles of neutron stars, in which the accreting matter is decelerated by radiation pressure. The physics of these systems is reviewed, with an emphasis on problems for which hydrodynamical simulations may be especially useful.