The Torelli group is the subgroup of the mapping class group of a surface acting trivially on the first homology of the surface. The first rational homology of the Torelli group is known for a closed surface of genus 2 by work of Mess, and for closed surfaces of genus at least 3 by work of Johnson. We will discuss forthcoming work with Putman that computes the second rational homology of the Torelli group for all closed surfaces of genus at least 6. In particular, we will show that this homology group is an algebraic representation of the symplectic group. Combined with the work of Kupers and Randal-Williams, this partially resolves Church and Farb's conjecture that the rational homology of the Torelli group is representation stable over the symplectic group.