Conjugate gradient type methods are one of the most important class of methods for solving large systems of linear equations. When those systems are obtained from the discretization of an ill-posed problem and the data are perturbed by noise, the iterations have to be arrested early in order to obtain a regularized solution. A widely used method in this case is the Conjugate Gradient method applied to the Normal Equations (CGNE), arrested with the discrepancy principle. Aim of this talk is to focus on the regularizing properties of the CGNE and to propose some heuristic and a-posteriori new stopping rules that improve the performance obtained by the discrepancy principle.