  In this paper we study Frobenius bimodules between noncommutative spaces
(quasi-schemes), developing some of their basic properties. If X and Y are
spaces, we study those Frobenius X,Y-bimodules M satisfying properties that are
natural in the context of noncommutative algebraic geometry, focusing in
particular on cartain "local" conditions on M. As applications, we prove
decomposition and gluing theorems for those Frobenius bimodules which have good
local properties. Additionally, when X and Y are schemes we relate Frobenius
X,Y-bimodules to the sheaf X,Y-bimodules introduced by Van den Bergh.
