This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that ''quantizing a system'' means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H-2(G, T)). The fundamental structural property is that there is a universal state space V embedding all states, a unitary representation space of the universal central extension G of G, on which the Pontryagin dual H-2(G,T)* of H-2(G,T) operates as a group of superselection rules, so that the superselection sectors are {V(h) subset-of C V\h is-an-element-of H-2(G,T)} and observables are self-adjoint operators V(h) --> V(h). This leads to a classification of time-evolution operators respecting the symmetries by H-2 (G, T) and a general understanding of ''anomalies'' as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, G coincides with GBAR, the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or ''mixed'' origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E(d) and the (connected) Poincare group P(d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals-in particular, the absence of ''fractional'' angular momentum in the plane is established and the difficulties specific to (2, 1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
