  We start by reviewing the formulation of noncommutative quantum mechanics as
a constrained system. Then, we address to the problem of field theories defined
on a noncommutative space-time manifold. The Moyal product is introduced and
the appearance of the UV/IR mechanism is exemplified. The emphasis is on
finding and analyzing noncommutative quantum field theories which are
renormalizable and free of nonintegrable infrared singularities. In this last
connection we give a detailed discussion of the quantization of the
noncommutative Wess-Zumino model as well as of its low energy behavior.
