The main aim of this paper is the construction of a smooth (sometimes called differential) extension hat{MU} of the cohomology theory complex cobordism MU, using cycles for hat{MU}(M) which are essentially proper maps Wto M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of Wto M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that hat{R}(M):=hat{MU}(M)otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.
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