Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I take to be algebraically closed).
It is classical that classifying normal affine toric varieties is the same thing as classifying rational polyhedral cones. The latter is a classical part of discrete and combinatorial geometry. (Apologies for overusing the word "classical".)
Question. Is there some combinatorial data which classifies non-normal affine toric varieties?
Background and what I am not looking for. I am not an algebraic geometer. I work a lot with non-commutative semigroups. So often people asked me what is the classification of finitely generated cancellative commutative monoids, expecting an answer like the group case. The category of affine monoids (=finitely generated submonoids of free abelian groups) is dual to the category of affine toric varieties, hence my question. The normal affine toric varieties correspond to saturated or normal affine monoids (also called integral polyhedral cones by some). For this reason I don't consider affine monoids as an answer to the question.