A projective spherical variety is a Mori dream space.
He further worked out the case of groups of type A and conjectured that combinatorial objects consisting of "homogeneous spherical data" classify spherical subgroups.
This classification is now complete according to Luna's program; see contributions of Bravi, Cupit-Foutou, Losev and Pezzini.
As conjectured by Knop, every "smooth" affine spherical variety is uniquely determined by its weight monoid.
Knop (2013) has been developing a program to classify spherical varieties in arbitrary characteristic.