This work is an extended version of Elements of Hydrodynamic Propulsion (Martinus Nijhoff 1984), and treats many subjects in a more general way or with more far-reaching results. The theories discussed are mainly linear although, where appropriate, nonlinearity is considered. The subjects treated have four different aspects: (1) basic hydrodynamics, in which topics such as non-divergent dipole flow, a general lifting surface theory, the orthogonality property of the flow far behind a screw propeller and the roll-up of free vortex sheets; (2) optimization theory of lifting surface systems which leave behind as little kinetic energy as possible under the constraint of prescribed force actions; (3) application of optimization theory to screw propellers with or without end plates at the tips of their blades and to sculling propulsion by means of wings; and (4) fundamental problems concerning the existence or non-existence of optimum propellers, especially in the case of unsteady propulsion. For naval architects and marine engineers whose work involves applying mathematics to propulsion problems, and applied mathematicians interested in hydrodynamics.