Combining classical results in univalent function theory with recent analogous results in higher dimensions, this text offers a unique overview of the field and details results leading to improvements in existence theorems for the Loewner differential equation in higher dimensions. Graham (mathematics, U. of Toronto) and Kohr (mathematics and computer science, Babes-Bolyai U., Romania) discuss compactness of the analog of the Carathéodory class in several variables, Loewner chains in several variables, linear-invariant families applied to the Euclidean unit ball and the polydisc, Bloch mappings, and infinite-dimensional theory of univalent mappings. The book concludes with a study of the Roper-Suffridge extension operator