the riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. this result is over a 100 years old, but the study and computation of such maps is still an active area. in this talk, i will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. as one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but i will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry. if time permits, i will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all new angles between 60 and 120 degrees. a closely related result states that any planar triangulation of n points can be refined by adding vertices and edges into a non-obtuse triangulation (no angles bigger than 90 degrees) in time $o(n^(5/2))$. no polynomial bound was previously known.