Abstract: The talk will consist of two parts. In the "group theory" part, we will show how to discretize an ordinary differential equation and obtain difference schemes invariant under the same Lie point symmetry group. In the "numerical" part, we will consider several second and third order nonlinear differential equations and show that the invariant schemes provide a much higher precision than standard schemes, without significantly increasing the the complexity of the computations. Moreover invariant schemes make it possible to treat singular solutions and to obtain numerical solutions valid up to and beyond singularities.