  In this paper we construct a bicategory of (super) algebra bundles over a
smooth manifold, where the 1-morphisms are bundles of bimodules. The main point
is that naive definitions of bimodule bundles will not lead to a well-defined
composition law in such a bicategory, at least not if non-invertible bimodules
and non-semisimple algebras are desired. This problem has not been addressed so
far in the literature. We develop a complete solution, and also address
symmetric monoidal structures as well as the corresponding questions of
dualizability.
