  In this work, I explore the concept of quantization as a mapping from
classical phase space functions to quantum operators. I discuss the early
history of this notion of quantization with emphasis on the works of
Schr\"odinger and Dirac, and how quantization fit into their overall
understanding of quantum theory in the 1920's. Dirac, in particular, proposed a
quantization map which should satisfy certain properties, including the
property that quantum commutators should be related to classical Poisson
brackets in a particular way. However, in 1946, Groenewold proved that Dirac's
mapping was inconsistent, making the problem of defining a rigorous
quantization map more elusive than originally expected. This result, known as
the Groenewold-Van Hove theorem, is not often discussed in physics texts, but
here I will give an account of the theorem and what it means for potential
"corrections" to Dirac's scheme. Other proposals for quantization have arisen
over the years, the first major one being that of Weyl in 1927, which was later
developed by many, including Groenewold, and which has since become known as
Weyl Quantization in the mathematical literature. Another, known as Geometric
Quantization, formulates quantization in differential-geometric terms by
appealing to the character of classical phase spaces as symplectic manifolds;
this approach began with the work of Souriau, Kostant, and Kirillov in the
1960's. I will describe these proposals for quantization and comment on their
relation to Dirac's original program. Along the way, the problem of operator
ordering and of quantizing in curvilinear coordinates will be described, since
these are natural questions that immediately present themselves when thinking
about quantization.
