We discuss cosmology of models with universal extra dimensions, where the Standard Model degrees of freedom live in a $4+n$ dimensional brane, with $n$ compact and small extra spatial dimensions. In these models, the simplest way to obtain the conventional 4-dimensional Planck scale starting with a low string scale is to have also some larger extra dimensions, where only gravity propagates. In such theories, dimensional reduction generically leads to at least two radion fields, one associated with the total volume of the extra spatial dimensions, and the other with the ratio of the sizes of small and large extra dimensions. In this paper, we discuss the impact of the radion fields on cosmology. We emphasize various aspects of radion physics such as radion coupling to the Standard Model fields, bare and dressed radion masses during inflation, dynamical stabilization of radions during and after inflation, radion decay life time and its late dominance in thermal history of the Universe as well as its quantum fluctuations during inflation. We argue that models where the radion plays the role of an inflaton or the inflaton is a brane scalar field, run into problems. We then present a successful inflation model with bulk scalar fields that seems to have all the desired properties. We also briefly discuss the possibility of radion as a cold dark matter candidate.