The Riemann Mapping Theorem guarantees that the upper half plane is conformallyequivalent to the interior domain determined by any polygon. Schwarz-Christoffel transformations provide explicit formulas for the maps that work. Popular textbook treatments of the topic range from motivational and contructive to proof-oriented. The aim of this paper is to combine the strengths of these expositions, filling in details and adding more information when necessary. In particular, careful attention is paid to the crucial fact, taken for granted in most elementary texts, that all conformal equivalences between the domains in question extend continuously to their closures.