Let Mn and Np be smooth manifolds, let f: Mn -» Np be smooth map, xeMn. Denote by TXM the tangent space of Mn at x, f induces a linear map dfx :TxM-»Tf(x)N. Recall that the rank of fat x is the dimension of the image of TXM under dfx. If the rank of f at x is less then min (n, p), then x is called a singular point of f and if the rank of f at x is less then p, then x is called a critical point of f. The set of critical points of f may not be a sub manifold of Mn in general. In this thesis we study some cases in which this set has the structure of a sub manifold. We also study the local behavior of a smooth map at a point in its domain. So we start by basic concepts and illustrative examples. We give the details of the proofs of known results.