  Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler,
structures on a compact manifold M, and $Diff_0$ the connected component of its
diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space
of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler
manifold M are cohomology classes which can be represented by a minimal
rational curve on a deformation of M. We determine the Teichmuller space of
hyperkahler structures on a hyperkahler manifold, identifying any of its
connected components with an open subset of the Grassmannian
$SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolov
positive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBM
classes. This is used to determine the Teichmuller space of symplectic
structures of Kahler type on a hyperkahler manifold of maximal holonomy. We
show that any connected component of this space is naturally identified with
the space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ is
the Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.
