We describe an algorithm for computing the conformal map between the unit disk and the interior on an n-gon to within accuracy \epsilon in time O(n), where the constant depends only on \epsilon (and is of order \log^2 \epsilon). The main idea is that there is a simple, fast, geometrically defined map which roughly approximates the conformal map with estimates independent of the domain. The uniform accuracy of the map comes from estimates in hyperbolic 3-dimensional geometry; it is fast to compute using results from computational geometry on the medial axis. This rough approximation can then be improved to any desired accuracy by a 2-phase process: first deform the given domain to a simplier one and then use a Newton type iteration based on solving the Beltrami equation using fast multipole techniques. This gives a good theoretical upper bound on the complexity of computing a conformal map, but has not been implemented.