  Quantitative estimates are obtained for the (finite) valence of functions
analytic in the unit disk with Schwarzian derivative that is bounded or of slow
growth. A harmonic mapping is shown to be uniformly locally univalent with
respect to the hyperbolic metric if and only if it has finite Schwarzian norm,
thus generalizing a result of B. Schwarz for analytic functions. A numerical
bound is obtained for the Schwarzian norms of univalent harmonic mappings.
