  We determine those smooth $n$--dimensional closed manifolds with $n \geq 4$
which admit round fold maps into ${\mathbb{R}}^{n-1}$, i.e.\ fold maps whose
critical value sets consist of disjoint spheres of dimension $n-2$ isotopic to
concentric spheres. We also classify such round fold maps up to $C^{\infty}$
$\mathcal{A}$--equivalence.
