  We describe a categorical framework for the classification of D-branes on
noncommutative spaces using techniques from bivariant K-theory of C*-algebras.
We present a new description of bivariant K-theory in terms of noncommutative
correspondences which is nicely adapted to the study of T-duality in open
string theory. We systematically use the diagram calculus for bivariant
K-theory as detailed in our previous paper. We explicitly work out our theory
for a number of examples of noncommutative manifolds.
