  We show how the motion through the extra dimensions of a gas of branes and
antibranes can, under certain circumstances, produce an era of inflation as
seen by observers trapped on a 3-brane, with the inflaton being the inter-brane
separation. Although most of our discussion refers to arbitrary p-branes, when
we need to be specific we assume that they are D-branes of Type II or Type I
string theory. For realistic brane couplings, such as those arising in string
theory, the inter-brane potentials are too steep to inflate the universe for
acceptably long times. However, for special regions of the parameter space of
brane-antibrane positions the brane motion is slow enough for there to be
sufficient inflation. Inflation would be more generic in models where the
inter-brane interactions are much weaker. The spectrum of primordial density
fluctuations predicted has index n slightly less than 1, and an acceptable
amplitude, provided that the extra dimensions have linear size 1/r ~ 10^{12}
GeV. Reheating occurs as in hybrid inflation, with the tachyonic instability of
the brane-antibrane system taking over for small separations. The tachyon field
can induce a cascade mechanism within which higher-dimension branes annihilate
into lower-dimension ones. We argue that such a cascade naturally stops with
the production of 3-branes in 10-dimensional string theory.
