We present the theory of chaos. We give the definition of chaos and some examles and counterexamples. We concentrate on the chaotic set of the smale horseshoe and we prove that it is chaotic through its topological conjugacy with sympolic dynamics i.e. the Bernoulli shift on the doubly infinite series of symbols. Finally we present the Melnikov theorem that proves chaos through the tranverse intersection of the stable and unstable manifolds of a saddle point in a Poincare map of a one and a half degrees of freedom periodic system of differential equations.