This dissertation compares several methods for computing conformal maps from sim-ply and multiply connected domains bounded by circles to target domains bounded by smooth curves and curves with corners. We discuss the use of explicit preliminary maps, including the osculation method of Grassmann to conformally map the target domain to a more nearly circular domain. The Fourier series method due to Fornberg and its generalizations to multiply connected domains are then applied to compute the maps to the nearly circular domains. The ?nal map is represented as a composition of the Fourier/Laurent series with the inverted explicit preliminary maps. A novel method for systematically re-moving corners with power maps is also implemented and composed with the Fornberg maps (which require smooth boundaries) and the level of error that can be expected when using Fourier series to treat domains with corners is illustrated. Some comparison to Wegmann's alternating projection method, which does not require smooth boundaries, is included. We also combine the Fornberg-like method with Karman-Tre?tz method for removing trailing edge corners in multi-element airfoils. The use of explicit maps has been suggested often in the past, but has rarely been carefully studied especially for the multiply connected case. A key contribution of this dissertation is the development of Matlab code for testing existing and new combinations of these various methods, in order to provide a tool for future applications, such as solving potential theory problems in general, multiply connected domains in the plane.