  We consider the cosmology that results if our observable universe is a
3-brane in a higher dimensional universe. In particular, we focus on the case
where our 3-brane is located at the $Z_2$ symmetry fixed plane of a $Z_2$
symmetric five-dimensional spacetime, as in the Ho\v{r}ava-Witten model
compactified on a Calabi-Yau manifold. As our first result, we find that there
can be substantial modifications to the standard Friedmann-Robertson-Walker
(FRW) cosmology; as a consequence, a large class of such models is
observationally inconsistent. In particular, any relationship between the
Hubble constant and the energy density on our brane is possible, including (but
not only) FRW. Generically, due to the existence of the bulk and the boundary
conditions on the orbifold fixed plane, the relationship is not FRW, and hence
cosmological constraints coming from big bang nucleosynthesis, structure
formation, and the age of the universe difficult to satisfy. We do wish to
point out, however, that some specific choices for the bulk stress-energy
tensor components do reproduce normal FRW cosmology on our brane, and we have
constructed an explicit example. As our second result, for a broad class of
models, we find a somewhat surprising fact: the stabilization of the radius of
the extra dimension and hence the four dimensional Planck mass requires
unrealistic fine-tuning of the equation of state on our 3-brane. In the last
third of the paper, we make remarks about causality and the horizon problem
that apply to {\it any} theory in which the volume of the extra dimension
determines the four-dimensional gravitational coupling. We point out that some
of the assumptions that lead to the usual inflationary requirements are
modified.
