Abstract: We discuss the discrete spectrum of Schroedinger operators with an attractive singular interaction supported by a curve or surface in R^n, n=2,3. Asymptotic expansions for the number of negative eigenvalues are derived as well as expansions for the j-th eigenvalue which is expressed in terms of a comparison operator with the potential determined by the geometry of the manifold. We also discuss the behaviour of band spectrum for periodic manifolds and list some open problems.