  This article is the author's contribution to the volume "Problems on mapping
class groups and related topics" which will be published in December 2006, with
Benson Farb as Editor. Various individuals were invited by Farb to submit open
problems which seemed interesting to them about surface mapping class groups.
We singled out a particular recent development, Hempel's concept of the
`Heegaard distance' of an element in the mapping class group, and used it to
describe a number of open questions which relate to the correspondence between
Heegaard surfaces in 3-manifolds and the mapping class group of a 2-manifold.
