  We introduce machinery to allow ``cut-and-paste''-style inductive arguments
in the Torelli subgroup of the mapping class group. In the past these arguments
have been problematic because restricting the Torelli group to subsurfaces
gives different groups depending on how the subsurfaces are embedded. We define
a category $\TSur$ whose objects are surfaces together with a decoration
restricting how they can be embedded into larger surfaces and whose morphisms
are embeddings which respect the decoration. There is a natural ``Torelli
functor'' on this category which extends the usual definition of the Torelli
group on a closed surface. Additionally, we prove an analogue of the Birman
exact sequence for the Torelli groups of surfaces with boundary and use the
action of the Torelli group on the complex of curves to find generators for the
Torelli group. For genus $g \geq 1$ only twists about (certain) separating
curves and bounding pairs are needed, while for genus $g=0$ a new type of
generator (a ``commutator of a simply intersecting pair'') is needed. As a
special case, our methods provide a new, more conceptual proof of the classical
result of Birman-Powell which says that the Torelli group on a closed surface
is generated by twists about separating curves and bounding pairs.
