Central in the topological approach to toric geometry  are several important spaces
which include  moment-angle complexes, the Davis-Januszkiewicz space and toric manifolds.
In any complex-oriented cohomology theory, the cohomology rings of many of these spaces
have elegant descriptions in terms of the underlying combinatorics. For KO-theory however the
situation is more complex. Even so, a surprising amount of the structure does survive from the
complex-oriented case. Necessary for the calculation is an explicit computation of the ring
structure of  KO-theory of an arbitrary product of infinite dimensional projective spaces. A
report of joint work with: Luis Astey, Martin Bendersky, Fred Cohen, Don Davis, Sam Gitler,
Mark Mahowald, Nigel Ray and Reg Wood.