The classical index theorem discovered by M. Atiyah and I. Singer was gradually generalized and extended ever since its formulation half a century ago. One possible line of generalizations tries to extend the index theorem to manifolds with singularities. The main aim of this book is to describe systematically such an extension in the case of a manifold with a suitable type of singularities. The first part of the book introduces elliptic operators on singular manifolds (i.e. on manifolds with conical singularities or on manifolds with edges). In the second part, the reader can find a review of the classical theory of pseudodifferential operators on manifolds and its extension to singular manifolds. A generalization of index theory to the case of singular manifolds is described in the third part of the book. The last part describes various additional topics and applications (Fourier integral operators on singular manifolds, relative elliptic theory, index theorems on manifolds with cylindrical ends, homotopy classifications of elliptic operators and the Lefschetz type formulae). Spectral flows and eta invariants are treated in two appendices. Manifolds with special types of singularities are appearing more and more often in various parts of mathematics. Hence the book will be very useful for mathematicians from different branches of mathematics and also for theoretical physicists.