  We define and study analogs of the Thouless charge pump for many-body gapped
systems in dimension $D$. We show how to attach a topological invariant to a
$D$-dimensional family of such systems, provided all of them have an on-site
$U(1)$ symmetry. For a large class of families we argue that this topological
invariant is an integer. In the case of gapped systems of free fermions in two
dimensions, the invariant can be expressed in terms of the curvature of the
Bloch-Berry connection. We also obtain a new formula for the Thouless charge
pump in 1d which involves only static linear response and is analogous to the
Streda formula for Hall conductivity.
