A rigorous study of stability of quantum systems, composed of many relativistic particles, under the influence of internal Coulomb forces and external force fields involves a careful spectral analysis of varied relativistic and quasi-relativistic self-adjoint operators of Hamiltonian type. This field of research is vast and dominated by mathematical physicists like E. Lieb, W. Thirring, I. Herbst, B. Simon, to mention a few. The present book is primarily addressed to mathematicians with an interest in the spectral analysis of operators of mathematical physics. Basic properties of Dirac, quasi-relativistic (so-called relativistic Hamiltonian operators), and Brown-Ravenhall operators with Coulomb potentials are presented. This is followed by an analysis of the nature of the spectrum of these operators, with a focus on the location of an essential spectrum and the existence of eigenvalues that are either isolated or embedded in the essential one. Various stability of matter results are discussed. The book comprises an outline of the thirty years of development of the subject. The literature is updated to the year 2010, concentrating mostly of the research done after 1992, with some emphasis on that conducted by the present authors.