Superstring theories contain massless (in the first instance) fields, referred to as the ‘dilaton’ and, more generally, ‘moduli’, whose effective potential is flat. A problem associated with the existence of the dilaton in particular is the need for it to settle into a minimum of the effective potential. If this occurs only with difficulty, there can be adverse effects on inflation. This problem is discussed in this chapter. On a slightly more positive note, the dilaton and moduli fields with their flat potentials before supersymmetry breaking are possible candidates for inflaton fields. There could be an era in the history of the universe during which all nine spatial dimensions are still of comparable size. The chapter addresses the cosmology of this era and how it joins on to the era with just three large spatial dimensions. It also addresses the question of why only three spatial dimensions become large. The cosmology of D-branes is also discussed.