In recent years derived categories of sheaves have come to be regarded as interesting geometric objects in their own right, and even as useful potential substitutes for varieties, spurred in part by developments in string theory (D-branes and homological mirror symmetry), noncommutative geometry and representation theory, and birational geometry. We will explore some of these ideas together, focusing on geometric structures on derived categories and the concrete ways to describe and compare them. The tentative plan is to follow the new book by Daniel Huybrechts on Fourier-Mukai Transforms in Algebraic Geometry, of which we'll have printouts (after taking solemn oaths not to distribute). In particular we'll learn how to find equivalences between derived categories and when we can reconstruct a variety from the derived category. Another main topic could be the description of derived categories as categories of modules for algebras. This relates to exceptional collections, quivers, Koszul duality and stems from a very concrete identification of derived categories of projective spaces (and more general Fano varieties). Other topics include McKay correspondence and noncommutative resolution of singularities, perverse sheaves and t-structures, and the geoemtric construction of flops. Some minimal familiarity with derived categories is helpful (eg Weibel's book) but we'll review everything. (Disclaimer: I am not a derived categorist, and hope to be learning a lot along with the group!)