Bridgeland stability conditions and birational geometry of moduli spaces

I will explain how one can use wall-crossing for Bridgeland stability conditions to deduce the most important aspects of the birational geometry of moduli spaces of sheaves on K3 surfaces. There is a piece-wise linear, almost surjective map from the space of stability conditions to the moveable cone of the moduli spaces; Bridgeland walls get mapped to walls in the moveable cone inducing flops or divisorial contractions. As applications, we obtain:
1. a description of the nef, moveable and effective cone of the moduli space in terms of the algebrai Mukai lattice on the K3, and
2. a proof of the Hassett-Tschinkel/Huybrechts/Sawon conjecture about the existence of Lagrangian fibrations for moduli of Gieseker-stable sheaves (or Bridgeland-stable complexes) on K3 surfaces.
This is based on joint work with Emanuele Macri.