  The goal of these lectures is to explain speaker's results on uniqueness
properties of spherical varieties. By a uniqueness property we mean the
following. Consider some special class of spherical varieties. Define some
combinatorial invariants for spherical varieties from this class. The problem
is to determine whether this set of invariants specifies a spherical variety in
this class uniquely (up to an isomorphism). We are interested in three classes:
smooth affine varieties, general affine varieties, and homogeneous spaces.
