A converse to the principle of contracting maps

In the paper we give an account of several versions of a converse to the principle of contracting maps. More exactly, we answer the question: under what conditions on an operator mapping a complete metric space into itself is there an equivalent metric in which the operator is contracting? We also consider the more general problem about the existence of an equivalent metric in which families and semigroups of operators are contracting, and we indicate connections of this problem with the theory of the stability of motion. A similar problem (about the existence of an equivalent norm) can be raised in the case of a Banach space.