In the early 1980s an elementary algorithm for computing conformal maps was discovered. Given points formula_22 in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve formula_23 with formula_24 This algorithm converges for Jordan regions in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a formula_25 curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the Loewner differential equation.