  This paper has 3 principal goals: (1) to survey what is know about mapping
class and Torelli groups of simply connected compact Kaehler manifolds, (2)
supplement these results, and (3) present a list of questions and open problems
to stimulate future work. Apart from reviewing general background, the paper
focuses on the case of hypersurfaces in projective space. We explain how older
results of Carlson--Toledo arXiv:alg-geom/9708002 and recent results of
Kreck--Su arXiv:2009.08054 imply that the homomorphism from the fundamental
group of the moduli space of hypersurfaces in P^4 to the mapping class group of
the underlying manifold has a very large kernel (contains a free group of rank
2) and has image of infinite index. This is in contrast to the case of curves,
where the homomorphism is an isomorphism.
