  It was recently pointed out that the fundamental Planck mass could be close
to the TeV scale with the observed weakness of gravity at long distances being
due the existence of new sub-millimeter spatial dimensions. In this picture the
standard model fields are localized to a $(3+1)$-dimensional wall or
``3-brane''. We show that in such theories there exist attractive models of
inflation that occur while the size of the new dimensions are still small. We
show that it is easy to produce the required number of efoldings, and further
that the density perturbations $\delta\rho/\rho$ as measured by COBE can be
easily reproduced, both in overall magnitude and in their approximately
scale-invariant spectrum. In the minimal approach, the inflaton field is just
the moduli describing the size of the internal dimensions, the role of the
inflationary potential being played by the stabilizing potential of the
internal space. We show that under quite general conditions, the inflationary
era is followed by an epoch of contraction of our world on the brane, while the
internal dimensions slowly expand to their stabilization radius. We find a set
of exact solutions which describe this behavior, generalizing the well-known
Kasner solutions. During this phase, the production of bulk gravitons remains
suppressed. The period of contraction is terminated by the blue-shifting of
Hawking radiation left on our wall at the end of the inflationary de Sitter
phase. The temperature to which this is reheated is consistent with the
normalcy bounds. We give a precise definition of the radion moduli problem.
