The mapping class group in genus g is the group of isotopy classes of orientation preserving homeomorphisms of a closed oriented surface of genus g. In variant forms one considers isotopy classes fixing a base point or of a surface with (connected) boundary. In any of these forms there is a homomorphism onto the symplectic group, Sp(2g), obtained by the action on the first homology group of the surface. The kernel of this homomorphism is known as the Torelli group in (genus g, and of the particular variant). This paper gives an informal, but informative, summary of the Torelli group - determining generators, describing its abelian quotients and other homogical information, explaining its connections to Teichmüller theory and 3-dimensional topology, and in the process raising many open questions.
For the entire collection see [Zbl 0512.00022].