A body irradiated by a sound field is known to experience a steady force that is called the acoustic radiation pressure. This force plays an important role in many physical phenomena, such as cavitation, sonoluminescence, acoustic levitation, etc. The existing theory of acoustic radiation pressure neglects dissipative effects. The present paper develops a theory that takes these effects into account, both dissipative mechanisms, viscous and thermal, being considered. It is shown that, when they are no longer negligible, the dissipative effects drastically change the radiation pressure. As a result, its magnitude and sign become different from those predicted by the ''classical'' theory neglecting losses.