  This thesis is dedicated to the study of K-theoretical properties of D-branes
and Ramond-Ramond fields. We construct abelian groups which define a homology
theory on the category of CW-complexes, and prove that this homology theory is
equivalent to the bordism representation of KO-homology, the dual theory to
KO-theory. We construct an isomorphism between our geometric representation and
the analytic representation of KO-homology, which induces a natural equivalence
of homology functors. We apply this framework to describe mathematical
properties of D-branes in type I String theory. We investigate the gauge theory
of Ramond-Ramond fields arising from type II String theory defined on global
orbifolds. We use the machinery of Bredon cohomology and the equivariant Chern
character to construct abelian groups which generalize the properties of
differential K-theory defined by Hopkins and Singer to the equivariant setting,
and can be considered as a differential extension of equivariant K-theory for
finite groups. We show that the Dirac quantization condition for Ramond-Ramond
fieldstrengths on a good orbifold is dictated by equivariant K-theory and the
equivariant Chern character, and study the group of flat Ramond-Ramond fields
in the particular case of linear orbifolds in terms of our orbifold
differential K-theory.
