As part of a series from the U. of Montreal promoting research in pure and applied mathematics, this volume constructs Riemann surfaces of infinite genus geometrically by pasting together plane domains and handles. In order to find a meaningful generalization of the classical theory of Riemann surfaces in the case of infinite genus, restrictions are imposed in terms of sizes and locations of handles, and in terms of gluing maps. The approach reveals information relevant to the classical theory of Riemann surfaces, the Torelli theorem, and the Kadomcev-Petviashvilli equations. The authors detail several important examples, including hyperelliptic surfaces of infinite genus, heat surfaces, and Fermi surfaces.