Abstract. 
We shall give an introduction to normal form theory of analytic
vector fields near a fixed point. We shall present classical results
concerning nonlinear perturbations of  linear (diagonal) vector fields.
Then, we shall extend this theory to perturbation X of more degenerate
"main part" X_0 (for instance,  a nilpotent linear part or quadratic (i.e
homogeneous of degree 2)  vector field). Finally, we shall address the
"rigidity problem" : under which condition, a pertubation X which is
formally conjugate to X_0, is, in fact, analytically conjugate to X_0 ?