The derived category of coherent sheaves, D^b(X), on an algebraic variety X is a homological object that encodes a lot of information about the geometry of X. The structure of D^b(X) is expected to encode subtle information about the birational geometry of X. For instance, if two varieties are related by a birational transformation that preserves the canonical bundle, the D-equivalence conjecture predicts that their derived categories are equivalent. I will give an overview of different approaches to the D-equivalence conjecture that have been successful in special cases. Then I will discuss a new general framework for studying the structure of derived categories using the space of Bridgeland stability conditions on D^b(X). I will formulate some conjectures about canonical flows on this space, which can be considered a categorical analog of the minimal model program in birational geometry, and explain how they imply the D-equivalence conjecture.