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In this paper we discuss some topics on the geometry of type II superstringbackgrounds with D-branes, in particular on the geometrical meaning of theD-brane charge, the Ramond-Ramond fields and the Wess-Zumino action. We seethat, depending on the behaviour of the D-brane on the four non-compactspace-time directions, we need different notions of homology and cohomology todiscuss the associated fields and charge: we give a mathematical definition ofsuch notions and show their physical applications. We then discuss the problemof corretly defining Wess-Zumino action using the theory of p-gerbes. Finally,we recall the so-called *-problem and make some brief remarks about it.

Topics on the geometry of D-brane charges and Ramond-Ramond fields