The Torelli subgroup of the mapping class group of a surface is the subgroup acting trivially on the first homology of the surface.  We will discuss recent joint work with Andrew Putman where we compute the second rational homology of the Torelli group for closed, orientable surfaces of genus at least 6.  Along the way we compute the first twisted homology of the Torelli group with coefficients in the abelianization of the maximal abelian cover of the closed surface.  We also prove some new purely representation theoretic results about certain infinitely presented representations of the symplectic group