Let N be the class of meromorphic functions f with the following properties: f has finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points of f are zeros of f′ and vice versa, with finitely many exceptions; f has finite order. It is proved that if f ∈ N, then f does not have wandering domains. Moreover, if f ∈ N and if ∞ is among the limit functions of fn in a cycle of periodic domains, then this cycle contains a singularity of f−1. (Here fn denotes the nth iterate of f) These results are applied to study Newton's method for entire functions g of the form where p and q are polynomials and where c is a constant. In this case, the Newton iteration function f(z) = z − g(z)/g′(z) is in N. It follows that fn(z) converges to zeros of g for all z in the Fatou set of f, if this is the case for all zeros z of g″. Some of the results can be extended to the relaxed Newton method.