A spherical homogeneous space is a homogeneous space \(G/H\), for a reductive group \(G\) such that a Borel subgroup of \(G\) has an open orbit on it.
Equivariant embeddings of such spaces have been studied extensively, especially in characteristic 0. The embeddings of a fixed spherical homogeneous space are, according to the Luna-Vust theory, described by coloured fans. This is a generalisation of the theory of toric varieties which are described by fans.
In this talk I want to discuss recent work about a well-behaved class of homogeneous spaces in characteristic \(p\) whose embeddings have nice properties.