  We prove that rational homology of the Torelli group of genus g is infinite
dimensional, provided g>6. This means that rational homology of the Torelli
space of genus g>6 is infinite dimensional. The Torelli groups with marked
points are also considered. In addition, we prove that rational homology of the
subgroup of the Torelli group of genus g generated by all the Dehn twists along
separating simple closed curves is infinite dimensional for g>2.
