  In this paper, we survey recent works on the structure of the mapping class
groups of surfaces mainly from the point of view of topology. We then discuss
several possible directions for future research. These include the relation
between the structure of the mapping class group and invariants of 3-manifolds,
the unstable cohomology of the moduli space of curves and Faber's conjecture,
cokernel of the Johnson homomorphisms and the Galois as well as other new
obstructions, cohomology of certain infinite dimensional Lie algebra and
characteristic classes of outer automorphism groups of free groups and the
secondary characteristic classes of surface bundles. We give some experimental
results concerning each of them and, partly based on them, we formulate several
conjectures and problems.
