  We investigate the semigroup of integer points inside a convex cone. We
extend classical results in integer linear programming to integer conic
programming. We show that the semigroup associated with nonpolyhedral cones can
sometimes have a notion of finite generating set. We show this is true for the
cone of positive semidefinite matrices (PSD) and the second-order cone (SOC).
Both cones have a finite generating set of integer points, similar in spirit to
Hilbert bases, under the action of a finitely generated group. We also extend
notions of total dual integrality, Gomory-Chv\'{a}tal closure, and
Carath\'{e}odory rank to integer points in arbitrary cones.
