  A Teichm\"uller space $Teich$ is a quotient of the space of all complex
structures on a given manifold $M$ by the connected components of the group of
diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group of
connected components of the diffeomorphism group. The moduli problems can be
understood as statements about the $\Gamma$-action on $Teich$. I will describe
the mapping class group and the Teichmuller space for a hyperkahler manifold.
It turns out that this action is ergodic. We use the ergodicity to show that a
hyperkahler manifold is never Kobayashi hyperbolic.
  This is my ICM submission, with review of some of my work on Teichmuller
spaces and moduli; proofs are sketched, new observations and some open problems
added.
