One of the central results in the subject of complex analysis, the Riemann mapping theorem unifies the areas of algebra, geometry, and topology. Today it is recognized as one of the most important theorems of nineteenth-century mathematics, even though its original proof by Georg Freidrich Bernhard Riemann was flawed. The search for generalizations and a better understanding of this theorem has continued throughout the twentieth century.

Early mapmakers, from Hipparchus in ancient Greece to Gerhardus Mercator in 16th-century Flanders, discovered that certain types of maps of the globe preserve shape information on a small scale. This means that the map accurately represents small circles as circles (rather than as ellipses or ovals), and that it accurately represents angles. Such maps are useful for navigation, even though they may grossly distort large-scale features. (For example, Greenland is disproportionately large in the Mercator projection.) A locally...