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This book was originally published in 1975. In Einstein's General Theory of Relativity the effects of gravitation are represented by the curvature of space-time. Physical processes occurring in the presence of gravitation must then be treated mathematically in terms of their behaviour in a curved space-time. One of the most basic of these processes is wave propagation, and this book gives a rigourous discussion of the local effects of curvature on the behaviour of waves. In the course of this discussion many techniques are developed which are also needed for a study of more general problems, in which the gravitational field itself plays a dynamical role. Although much of the book deals with four-dimensional space-time, the n-dimensional case is also treated, more briefly. The subject-matter is also of interest in other branches of mathematical physics and, as a fresh account of the classical work of Hadamard and M. Riesz, in the theory of partial differential equations.