Classification of $G$-varieties of complexity 1

We consider the problem of finding a combinatorial description of the algebraic varieties in a given birational class that admit an action of a reductive group $G$. This is a direct generalization of the theory of toric varieties. A general approach to this problem is described, and the solution is given for varieties in which the orbits in general position of a Borel subgroup $G$ have codimension 1 (varieties of complexity 1).