  In this paper, we consider the normal form problem of a commutative family of
germs of diffeomorphisms at a fixed point, say the origin, of $\mathbb{K}^n$
($\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$). We define a notion of integrability
of such a family. We give sufficient conditions which ensure that such an
integrable family can be transformed into a normal form by an analytic (resp. a
smooth) transformation if the initial diffeomorphisms are analytic (resp.
smooth).
