In this talk we use ideas from algebraic geometry to propose a method for solving the 0-1 feasibility problem for integer programming. The method systematically enumerates all feasible solutions or shows that none exists, gives structural information on the set of feasible solutions and characterizes their number. It provides a natural generalization of Farkas lemma, leads to a strong duality theory for these problems in the sense that it provides a certificate of infeasibility and gives rise to a way of performing sensitivity analysis for integer programming. We also provide several examples that offer insights on the method and its properties. Finally, we illustrate how the method extends to general integer optimization problems.