Problems and algorithms for affine semigroups

In this article we overview those aspects of the theory of affine semigroups and their algebras that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the further development of the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different ``sizes'' of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects -- polytopal semigroups and algebras -- provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting. Our counterexample to certain covering conjectures was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.