We establish axiomatic characterizations of $K$-theory and $KK$-theory for
real C*-algebras. In particular, let $F$ be an abelian group-valued functor on
separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable,
and split exact, then $F$ factors through the category $KK$. Also, if $F$ is
homotopy invariant, stable, half exact, continuous, and satisfies an
appropriate dimension axiom, then there is a natural isomorphism $K(A) \to
F(A)$ for a large class of separable real C*-algebras $A$. Furthermore, we
prove that a natural transformation $F(A) \to G(A)$ of homotopy invariant,
stable, half-exact functors which is an isomorphism when $A$ is complex is
necessarily an isomorphism when $A$ is real.