This monograph deals with parametric minimal surfaces in Euclidean space. The author presents a broad survey which extends from the classical beginnings to the current situation whilst highlighting many of the subject's main features and interspersing the mathematical development with pertinent historical remarks. The presentation is complete and is complemented by a bibliography of nearly 1600 references. The careful expository style and emphasis on geometric aspects ensure that the work can be used for graduate level courses in mathematics. Moreover, in recent years the theory of minimal surfaces is finding increasing application to other areas of mathematics and the physical sciences so this account will appeal to non-specialists as well. This, the first of the proposed two volumes, is a revised and updated translation of the initial five chapters of Vorlesungen uber Minimalflachen, augmented by eight relevant appendices and a collection of open problems. It provides a thorough, self-contained introduction to minimal surfaces.".