Conditions are given for a l map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton's method. For finitely differentiable maps and functions, these conditions can only be necessary, but in the smooth case, i.e. for l=∞, they are also sufficient. The characterization rests upon the structure of the fixed point set of T, and it is best possible as is demonstrated through examples.