In the talk we will outline a proof of the convergence of normal form power series for suitably nonsingular submanifolds under the action of a broad class of infinite-dimensional Lie pseudo-groups. The construction of normal forms relies on the method of equivariant moving frames, while the convergence proof is based on the realization that a normal form is the solution to an initial value problem for an involutive system of differential equations, whose analyticity is guaranteed by the Cartan–Kähler theorem. Our theorem includes, as a particular example, Chern and Moser’s celebrated convergence theorem of normal forms of real hypersurfaces.