This book presents the proceedings of a conference held in celebration of the 50th anniversary of the introduction of the Initial Mass Function (IMF) concept by Ed Salpeter along with Ed Salpeter's 80th birthday. It provides a vast amount of information, and updates the Herstmonceux book on the IMF.This volume goes far beyond a regular proceedings in that it presents a large collection of long reviews on IMF determinations in the Galaxy, in galaxies, and in the Early Universe. In addition, the book includes reviews on the deep implications of the IMF on star formation theories, and on the physical conditions of the gas before and after star formation. As a consequence, the reader will find a lot of basic information needed for graduate courses on Stellar Evolution, Star Formation, Interstellar Medium, Galactic Dynamics, Formation and Evolution of Galaxies.The book covers all aspects of the current research in this field, as testified by the large number of contributing scientists.