This volume contains articles that document the advances in the subject of total positivity since the 1970s. The material is divided into ten chapters. While some of the articles are of a survey nature, others present new results. Also, some papers contain introductory material and are therefore accessible to non-experts interested in becoming familiar with the important ideas and techniques of total positivity. This book should be of value to mathematicians, engineers and computer scientists whose work involves applications of total positivity to problems in the theory of spline functions, numerical quadrature, nonlinear analysis, entire functions, probability, mathematical biology, statistics, approximation theory, combinatorics, geometric modelling, matrix theory and integral equations.