Modern ideas are presented concerning the source of an earthquake and the seismotectonic source of a tsunami. The main physical processes taking place at a tsunami source are described. Estimation is performed of the role of secondary effects: displacements of the bottom, occurring in its own plane, Coriolis force, and density stratification of the water. The Okada formulae are presented, and the technique is exposed for calculating residual bottom deformations caused by an underwater earthquake. Within the framework of linear potential theory of an incompressible liquid in a basin of fixed depth, the general analytical solution is constructed for the two-dimensional (2D) and three-dimensional (3D) problems of tsunami generation by bottom deformations of small amplitudes. The solution of the 3D problem is constructed in both Cartesian and cylindrical coordinates. For a series of model bottom deformation laws (piston, membrane and running displacements, bottom oscillations and alternating-sign displacement) physical regularities are revealed that relate the amplitude, energy and direction of tsunami wave emission to peculiarities of the bottom deformation at the source. In some cases, the theoretical regularities, obtained within potential theory, are compared with dependences following from the linear theory of long waves and, also, with the results of laboratory experiments.