In topology the mapping class group of a surface, (sometimes called
the modular group or Teichmüller modular group), is the group of
homeomorphisms of the surface viewed up to continuous deformation. It
is of fundamental importance for the study of 3-manifolds and various
other areas of mathematics. I'll give a gentle introduction to the
classification of mapping classes which will be used as an excuse to
tramp around in various areas of low-dimensional topology.