  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_2-3). 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 $SO(b_2-3, 3)/SO(2)\times
SO(b_2 -3, 1)$. We use this result to obtain a Torelli theorem identifying any
connected component of birational moduli space with a quotient of a period
space by an arithmetic subgroup. When $M$ is a Hilbert scheme of $n$ points on
a K3 surface, with $n-1$ a prime power, our Torelli theorem implies the usual
Hodge-theoretic birational Torelli theorem (for other examples of hyperkahler
manifolds the Hodge-theoretic Torelli theorem is known to be false).
