  In this review we study BPS D-branes on Calabi-Yau threefolds. Such D-branes
naturally divide into two sets called A-branes and B-branes which are most
easily understood from topological field theory. The main aim of this paper is
to provide a self-contained guide to the derived category approach to B-branes
and the idea of Pi-stability. We argue that this mathematical machinery is hard
to avoid for a proper understanding of B-branes. A-branes and B-branes are
related in a very complicated and interesting way which ties in with the
``homological mirror symmetry'' conjecture of Kontsevich. We motivate and
exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops
and orbifolds are discussed at some length. In the latter case we describe the
role of McKay quivers in the context of D-branes. These notes are to be
submitted to the proceedings of TASI03.
