  These notes are a writeup of lectures given at the twelfth Oporto meeting on
``Geometry, Topology, and Physics,'' and at the Adelaide workshop ``Strings and
Mathematics 2003,'' primarily geared towards a physics audience. We review
current work relating boundary states in the open string B model on Calabi-Yau
manifolds to sheaves. Such relationships provide us with a mechanism for
counting open string states in situations where the physical spectrum
calculation is nearly intractable -- after translating to mathematics, such
calculations become easy. We describe several different approaches to these
models, and also describe how these models are changed by varying physical
circumstances -- flat B field backgrounds, orbifolds, and nonzero Higgs vevs.
We also discuss mathematical interpretations of operator products, and how such
mathematical interpretations can be checked physically. One of the motivations
for this work is to understand the precise physical relationship between
boundary states in the open string B model and derived categories in
mathematics, and we outline what is currently known of the relationship.
