  Present state of the study of nonlinear ``integrable" systems related to the
group of area-preserving diffeomorphisms on various surfaces is overviewed.
Roles of area-preserving diffeomorphisms in 4-d self-dual gravity are reviewed.
Recent progress in new members of this family, the SDiff(2) KP and Toda
hierarchies, is reported. The group of area-preserving diffeomorphisms on a
cylinder plays a key role just as the infinite matrix group GL($\infty$) does
in the ordinary KP and Toda lattice hierarchies. The notion of tau functions is
also shown to persist in these hierarchies, and gives rise to a central
extension of the corresponding Lie algebra.
