Given a metric space (X, d), there are several notions of it being negatively curved. In this talk, we single out certain consequences of negative curvature – which are themselves weaker than (X, d) being negatively curved – that turn out to be very useful in proving results about holomorphic maps. We shall illustrate what this means by giving a proof of the Wolff-Denjoy theorem. (This theorem says that given a holomorphic self-map f of the open unit disc, either f has a fixed point in the open unit disc or there exists a point p on the unit circle such that ALL orbits under the successive iterates of f approach p.) For most of this talk, we shall focus on metric spaces or on geometry in one complex variable. Towards the end, we shall briefly point out what can be proved in domains in higher dimensions that have the aforementioned negative-curvature-type properties. This part of the talk is joint work with Andrew Zimmer.

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