  We give complete and exact descriptions of spaces of ultradifferentiable
functions that are closed under composition with either holomorphic or
ultradifferentiable functions -- which are two distinct cases. The proof works
by considering formal power series, and stability under differentiation is not
required. As an application of the power series approach we reprove regularity
results for solutions of ode's and pde's without employing tedious estimates
imploying the Fa\`a di Bruno formula for higher derivatives of composite maps.
