Spherical variety

In algebraic geometry, given a reductive algebraic group "G" and a Borel subgroup "B", a spherical variety is a "G"-variety with an open dense "B"-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties.
There is also a notion of real spherical varieties.
A projective spherical variety is a Mori dream space.
Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory.
In his seminal paper, develops a framework to classify complex spherical subgroups of reductive groups; he reduces the classification of spherical subgroups to wonderful subgroups. He works out completely the case of groups of type A and conjectures that the combinatorial objects (homogeneous spherical data) he introduces indeed provide a combinatorial classification of spherical subgroups. This has been known as the Luna Conjecture.
This classification is now completed 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. 
This uniqueness result has been proved by Losev.
 has been developing a program to classify spherical varieties in arbitrary characteristic.