  We review various aspects of the topological classification of D-brane
charges in K-theory, focusing on techniques from geometric K-homology and
Kasparov's KK-theory. The latter formulation enables an elaborate description
of D-brane charge on large classes of noncommutative spaces, and a refined
characterization of open string T-duality in terms of correspondences and
KK-equivalence. The examples of D-branes on noncommutative Riemann surfaces and
in constant H-flux backgrounds are treated in detail. Mathematical
constructions include noncommutative generalizations of Poincare duality and
K-orientation, characteristic classes, and the Riemann-Roch theorem.
