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Synopsis This book is based on a course of lectures given to PhD students at the Delhi Centre of the Indian Statistical Institute during the years 1980-1985. The approach is inspired by the lectures of G.W. Mackey and V.S. Varadarajan and brings out the role of group representations as the main thread passing through some major results like Gleason's theorem on the characterization of quantum states, Wigner's Unitarity-antiunitarity theorem on the automorphisms for the lattice of orthogonal projections in a Hilbert space, Mackey's Imprimitivity theorem for quantum systems with a configuration observable, and the rise of important and physically meaningful observables through the infinitesimal generators of projective unitary representations of the Galilean and the inhomogeneous Lorentz groups.