  The classical trace map is a highly non-trivial map from algebraic K-theory
to topological Hochschild homology (or topological cyclic homology) introduced
by B\"okstedt, Hsiang and Madsen. It led to many computations of algebraic
K-theory of rings. Hesselholt and Madsen recently introduced a Z/2-equivariant
version of Waldhausen S-construction for categories with duality. The output is
a certain spectrum with involution, called the real K-theory spectrum KR, and
associated bigraded groups analogous to Atiyah's real (topological) K-groups.
This thesis develops a theory of topological Hochschild homology for categories
with duality, and a Z/2-equivariant trace map from real K-theory to it. The
main result of the thesis is that stable KR of the category of projective
modules over a split square zero extension of a ring is equivalent to the real
topological Hochschild homology of the ring with appropriate coefficients. This
is the real version a theorem of Dundas-McCarthy for ordinary K-theory.
