  One approach to multivariate operator theory involves concepts and techniques
from algebraic and complex geometry and is formulated in terms of Hilbert
modules. In these notes we provide an introduction to this approach including
many proofs. We are particularly interested in examples related to hermitian
holomorphic vector bundles and we study submodules and reducing submodules in
such cases. We go into some detail concerning a problem of Zhu on the reducing
subspaces of powers of the Bergman shift as well as more recent work of the
author and J. Sarkar on proper submodules which are unitarily equivalent to the
orginal. Although the basic results are not new, there is some novelty in the
details and the organization of the material.
