  We study the circumstances under which a Kaluza-Klein reduction on an
n-sphere, with a massless truncation that includes all the Yang-Mills fields of
SO(n+1), can be consistent at the full non-linear level. We take as the
starting point a theory comprising a p-form field strength and (possibly) a
dilaton, coupled to gravity in the higher dimension D. We show that aside from
the previously-studied cases with (D,p)=(11,4) and (10,5) (associated with the
S^4 and S^7 reductions of D=11 supergravity, and the S^5 reduction of type IIB
supergravity), the only other possibilities that allow consistent reductions
are for p=2, reduced on S^2, and for p=3, reduced on S^3 or S^{D-3}. We
construct the fully non-linear Kaluza-Klein Ansatze in all these cases. In
particular, we obtain D=3, N=8, SO(8) and D=7, N=2, SO(4) gauged supergravities
from S^7 and S^3 reductions of N=1 supergravity in D=10.
