  Let G be a complex connected reductive group. I. Losev has shown that a
smooth affine spherical G-variety X is uniquely determined by its weight
monoid, which is the set of irreducible representations of G that occur in the
coordinate ring of X. In this paper we use a combinatorial characterization of
the weight monoids of smooth affine spherical varieties to classify: (a) all
such varieties when G is $\mathrm{SL}(2) \times \mathbb{C}^{\times}$ and (b)
all such varieties for G simple which have a G-saturated weight monoid of full
rank. We also use the characterization and F. Knop's classification theorem for
multiplicity free Hamiltonian manifolds to give a new proof of C. Woodward's
result that every reflective Delzant polytope is the moment polytope of such a
manifold.
