  We prove that any D-dimensional theory comprising gravity, an antisymmetric
n-index field strength and a dilaton can be consistently reduced on S^n in a
truncation in which just $n$ scalar fields and the metric are retained in
(D-n)-dimensions, provided only that the strength of the couping of the dilaton
to the field strength is appropriately chosen. A consistent reduction can then
be performed for n\le 5; with D being arbitrary when n\le 3, whilst D\le 11 for
n=4 and D\le 10 for n=5. (Or, by Hodge dualisation, $n$ can be replaced by
(D-n) in these conditions.) We obtain the lower dimensional scalar potentials
and construct associated domain wall solutions. We use the consistent reduction
Ansatz to lift domain-wall solutions in the (D-n)-dimensional theory back to D
dimensions, where we show that they become certain continuous distributions of
(D-n-2)-branes. We also examine the spectrum for a minimally-coupled scalar
field in the domain-wall background, showing that it has a universal structure
characterised completely by the dimension n of the compactifying sphere.
