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