In this habilitation thesis, we deal with explicit and implicit representations of lattice point sets via integral bases and via short rational generating functions. As main results this includes, - new approaches to compute Hilbert bases of cones and Graver bases of lattices, - computation of Graver bases exploiting existing symmetry, - extension of the notion of a test set to a certain class of convex integer programming problems, - polynomial time algorithm for linear optimization over the lattice points of 3-dimensional transportation polytope when two dimensions are kept fix, - FPTAS for maximizing a nonnegative polynomial over the (mixed-) integer points of a polytope, - polynomial size encoding of Groebner bases of toric ideals as short rational generating functions.
