The question addressed in this monograph is the relationship between the timereversible Newton dynamics for a system of particles interacting via elastic collisions and the irreversible Boltzmann dynamics which gives a statistical description of the collision mechanism. Two types of elastic collisions are considered: hard spheres and compactly supported potentials. Following the steps suggested by Lanford in 1974, the authors describe the transition from Newton to Boltzmann by proving a rigorous convergence result in short time, as the number of particles tends to infinity and their size simultaneously goes to zero, in the BoltzmannGrad scaling. Boltzmann's kinetic theory rests on the assumption that particle independence is propagated by the dynamics. This assumption is central to the issue of appearance of irreversibility. For finite numbers of particles, correlations are generated by collisions. The convergence proof establishes that for initially independent configurations, independence is statistically recovered in the limit. This book is intended for mathematicians working in the fields of partial differential equations and mathematical physics and is accessible to graduate students with a background in analysis. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in partial differential equations and mathematical physics. Table of Contents I. Introduction  The low density limit
 The Boltzmann equation
 Main results
II. The case of hard spheres  Microscopic dynamics and BBGKY hierarchy
 Uniform a priori estimates for the BBGKY and Boltzmann hierarchies
 Statement of the convergence result
 Strategy of the proof of convergence
III. The case of shortrange potentials  Twoparticle interactions
 Truncated marginals and the BBGKY hierarchy
 Cluster estimates and uniform a priori estimates
 Convergence result and strategy of proof
IV. Termwise convergence  Elimination of recollisions
 Truncated collision integrals
 Proof of convergence
 Concluding remarks
 Bibliography
 Index
