  In this article we give an introduction to the Fock quantization of the
Maxwell field. At the classical level, we treat the theory in both the
covariant and canonical phase space formalisms. The approach is general since
we consider arbitrary (globally-hyperbolic) space-times. The Fock quantization
is shown to be equivalent to the definition of a complex structure on the
classical phase space. As examples, we consider stationary space-times as well
as ordinary Minkowski space-time. The account is pedagogical in spirit and is
tailored to beginning graduate students. The paper is selfcontained and is
intended to fill an existing gap in the literature.
