We provide an axiomatic framework for the study of smooth extensions of
generalized cohomology theories. Our main results are about the uniqeness of
smooth extensions, and the identification of the flat theory with the
R/Z-theory.

In particular, we show that there is a unique smooth extension of K-theory
and of MU-cobordism with a unique multiplication, and that the flat theory in
these cases is naturally isomorphic to the homotopy theorist's version of the
cohomology theory with R/Z-coefficients. For this we only require a small set
of natural compatibility conditions.