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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 itsdiffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller spaceof symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahlermanifold M are cohomology classes which can be represented by a minimalrational curve on a deformation of M. We determine the Teichmuller space ofhyperkahler structures on a hyperkahler manifold, identifying any of itsconnected components with an open subset of the Grassmannian$SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolovpositive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBMclasses. This is used to determine the Teichmuller space of symplecticstructures of Kahler type on a hyperkahler manifold of maximal holonomy. Weshow that any connected component of this space is naturally identified withthe space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ isthe Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.

Teichmuller space for hyperkahler and symplectic structures