Noncommutative functions, introduced in the early ’70s by J.L. Taylor in the context of a functional calculus for nonncommuting operators, can be viewed from a modern perspective as a function theory on operator spaces. In this talk, we show that certain noncommutative domains can be endowed with pseudometrics that arise naturally from their very structure, and with respect to which noncommutative analytic maps are contractions. We will conclude with some applications. The talk is based on joint work with V. Vinnikov.