The mapping class group of an oriented manifold M is the group of connected components of its group of orientation preserving diffeomorphisms. When M is a closed surface of genus g, there is a
very close (and classical) relation between the mapping class group and the moduli space of compact Riemann surfaces of genus g that comes from Teichmuller theory. One manifestation of this relationship is that the (orbifold) fundamental group of the moduli space is isomorphic to the mapping class group of the surface.
One can ask if this relation persists in higher dimensions. Surprisingly, very little is known and the subject is in its infancy. In this talk, after reviewing the classical case, I will discuss the problem of understanding mapping class groups of simply connected complex projective manifolds and then specialize to the case of hypersurfaces in projective (n+1)-space when n>2. The most definitive results are due to Kreck and Su and apply when n=3. I will also explain why the mapping class group of a 3-dimensional hypersurface of degree > 3 is not isomorphic to the fundamental group of the corresponding moduli space, a consequence of the work of Kreck-Su and older work of Carlson and Toledo.