Following some motivating remarks on the role of group theory in physics, it is explained how a systematic study of the asymptotic symmetry group of general relativity, the Bondi-Metzner-Sachs group, can give some very general insights into the nature of the energy, momentum and angular momentum structure of curved (asymptotically flat) space-times, as well as of quantum theory in these space-times. In particular, a classical 'Bondi angular momentum' vector is defined for any B-invariant classical system. However, the representations which arise in any fundamental study of this (or any other) quantum theory are projective rather than linear. This paper is devoted to the problem of showing that each continuous projective representation arises from (that is, can be 'lifted' to) a continuous linear (in fact, unitary) representation of the universal covering group. An almost complete proof of this is given for the two very different cases in which the group is given a Hilbert or nuclear topology. This reduces the study of the group in quantum theory to an ordinary unitary representation problem which has already been studied, and implies, for example, that nothing more general than 'two-valued representations' need be considered. Some remarks are made on the physical interpretation of the known irreducibles, and on outstanding problems of this representation theory.