The spectral theory of Schr?dinger operators, in particular those with random potentials, continues to be a very active field of research. This work focuses on various known criteria in the spectral theory of selfadjoint operators in order to identify the spectrum and its components ? la Lebesgue decomposition. Key features and topics: * Well-developed exposition of criteria that are especially useful in determining the spectra of deterministic and random Schr?dinger operators occurring in quantum theory * Systematically uses measures and their transforms (Fourier, Borel, wavelet) to present a unifying theme * Establishes criteria for identifying the spectrum * Examines a series of applications to show point spectrum and continuous spectrum in some models of random operators * Presents a series of spectral-theoretic results for the perturbed operators introduced in the earlier chapters with examples of localization and delocalization in the theory of disordered systems * Presents modern criteria (using wavelet transform, eigenfunction decay) that could be used to do spectral theory * Unique work in book form combining the presentation of the deterministic and random cases, which will serve as a platform for further research activities This concise unified presentation is aimed at graduate students and researchers working in the spectral theory of Schr?dinger operators with either fixed or random potentials in particular. However, given the large gap that this book fills in the literature, it will serve a wider audience of mathematical physicists in its contribution to works in spectral theory.