Transition radiation is emitted when physical objects (particles, electromagnetic pulses, etc.) that have no eigenfrequencies move uniformly through inhomogeneous or nonstationary media. The effect is kinematic in character and is observed for waves of any physical nature. This review paper is devoted to the current state of the theory of acoustic transition radiation and transition scattering in media whose motion can be described by the hydrodynamic equations. The most characteristic problems in the theory of acoustic transition radiation are discussed, including the crossing of separation boundaries by a source, emission in a nonstationary medium, and so on. A number of diffraction problems of the theory are examined (e.g., the emission of sound by a source moving near inhomogeneities of different type). The last section is devoted to acoustic transition scattering. Particular attention is devoted to methodological aspects.