  Let $M$ be an irreducible holomorphically symplectic manifold. We show that
all faces of the Kahler cone of $M$ are hyperplanes $H_i$ orthogonal to certain
homology classes, called monodromy birationally minimal (MBM) classes.
Moreover, the Kahler cone is a connected component of a complement of the
positive cone to the union of all $H_i$. We provide several characterizations
of the MBM-classes. We show the invariance of MBM property by deformations, as
long as the class in question stays of type (1,1). For hyperkahler manifolds
with Picard group generated by a negative class $z$, we prove that $\pm z$ is
Q-effective if and only if it is an MBM class. We also prove some results
towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.
