A mapping class group of an
oriented manifold is a quotient of its diffeomorphism
group by the isotopies. We compute a mapping class group
of a hypekahler manifold $M$, showing that it is
commensurable to an arithmetic subgroup in SO(3, b_23). A
Teichmuller space of $M$ is a space of complex structures
on $M$ up to isotopies. We define a birational Teichmuller
space by identifying certain points corresponding to
bimeromorphically equivalent manifolds, and show that the
period map gives an isomorphism of the birational
Teichmuller space and the corresponding period space
$SO(b_23, 3)/SO(2)\times SO(b_2 3, 1)$. We use this
result to obtain a Torelli theorem identifying any
connected component of the birational moduli space with a
quotient of a period space by an arithmetic subgroup.
