In the first part of this talk we introduce integer programming models and the two ingredients for their practical solution, namely, the simplex and the branch-and-bound algorithms. Next, we show how to formulate the Traveling Salesman problem, probably the most famous problem in combinatorial optimization, as an integer program having an exponential number of inequalities. Despite the huge size of the resulting formulation, we present a solution approach where only a small subset of inequalities has to be explicitly considered for computing an optimal solution.