In this thesis we introduce the Laplace-Beltrami operator on Riemannian manifolds and study its spectrum on noncompact manifolds. For compact connected manifolds it is known that the Laplace-Beltrami operator has a discrete spectrum consisting of real spectral points, but in the case of noncompact manifolds the picture becomes more complicated. The manifolds Rn and Hn, whose spectra are shown to be [0,∞] and [(n - 1)2/4,∞) respectively, serve as examples of this. We also find that the spectrum of the wave operator, that is the Laplace-Beltrami operator on Minkowski space, is R. In order to calculate these spectra we present the notion of approximate eigenvalue sequences and clarify how these sequences relate to the spectrum of a linear operator.