  Quantization is not a straightforward proposition, as shown by Groenewold's
and Van Hove's discovery, more than fifty years ago, of an "obstruction" to
quantization. Their "no-go theorems" assert that it is impossible to
consistently quantize every classical polynomial observable on the phase space
R^{2n} in a physically meaningful way. Similar obstructions have been found for
S^2 and T*S^1, buttressing the common belief that no-go theorems should hold in
some generality. Surprisingly, this is not so--it has been proven that there
are no obstructions to quantizing either T^2 or T*R_+. In this paper we work
towards delineating the circumstances under which such obstructions will
appear, and understanding the mechanisms which produce them. Our objectives are
to conjecture--and in some cases prove--generalized Groenewold-Van Hove
theorems, and to determine the maximal Lie subalgebras of observables which can
be consistently quantized. This requires a study of the structure of Poisson
algebras and their representations. To these ends we include an exposition of
both prequantization (in an extended sense) and quantization theory, here
formulated in terms of "basic algebras of observables." We then review in
detail the known results for R^{2n}, S^2, T*S^1, T^2, and T*R_+, as well as
recent theoretical work.
