This is a very interesting, nicely written book on deep mathematical analogies between hydrodynamics, geometric optics, and mechanics. The author shows that the concept of a vortex has a very long and extremely rich history - and also modern developments - in natural sciences and mathematics. After a historical introduction (Descartes, Leibniz, Newton, Bernoulli, Voltaire, Maupertuis, Clairaut, Helmholtz, Thompson,...), Chapter 1 discusses hydrodynamics, geometric optics, and classical mechanics. A general vortex theory is discussed in Chapter 2 and geodesics on Lie groups with a left-invariant metric in Chapter 3. The last chapter includes a discussion of the vortex method for integrating Hamilton equations. In the supplements, the reader can find various topics (vorticity invariants and secondary hydrodynamics, quantum mechanics and hydrodynamics, vortex theory of adiabatic equilibrium processes). The book will be surely of great interest to researchers and postgraduate students in mathematical physics and mechanics.