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We use wall-crossing with respect to Bridgeland stability conditions tosystematically study the birational geometry of a moduli space M of stablesheaves on a K3 surface X:  1. We describe the nef cone, the movable cone, and the effective cone of M interms of the Mukai lattice of X.  2. We establish a long-standing conjecture that predicts the existence of abirational Lagrangian fibration on M whenever M admits an integral divisorclass D of square zero (with respect to the Beauville-Bogomolov form).  These results are proved using a natural map from the space of Bridgelandstability conditions Stab(X) to the cone Mov(X) of movable divisors on M; thismap relates wall-crossing in Stab(X) to birational transformations of M. Inparticular, every minimal model of M appears as a moduli space ofBridgeland-stable objects on X.

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations