This book provides a comprehensive and self-contained exposition of the algebro-geometric theory of singularities of plane curves, covering both its classical and its modern aspects. The book gives a unified treatment, with complete proofs, presenting modern results which have only ever appeared in research papers. It updates and correctly proves a number of important classical results for which there was formerly no suitable reference, and includes new, previously unpublished results as well as applications to algebra and algebraic geometry. This book will be useful as a reference text for researchers in the field. It is also suitable as a textbook for postgraduate courses on singularities, or as a supplementary text for courses on algebraic geometry (algebraic curves) or commutative algebra (valuations, complete ideals).