This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry.  It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions.  Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families.

This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.

F. Bogomolov is Professor at the Courant Institute, NYU. He is best known for his pioneering work on hyperkähler manifolds. B. Hassett is Professor and Chair of the department of Mathematics at Rice University. He published two books and around 50 papers on Algebraic and Arithmetic Geometry. Yuri Tschinkel is Professor at the Courant Institute, NYU and Director of the Mathematics and the Physical Sciences Division at the Simons Foundation.

Foreword.- Introduction.- A. Bertram and I. Coskun, The birational geometry of the Hilbert scheme of points on surfaces.- F. Bogomolov and Ch. Böhning, Isoclinism and stable cohomology of wreath products.- F. Bogomolov, I. Karzhemanov, and K. Kuyumzhiyan, Unirationality and existence of infinitely transitive models.- I. Cheltsov, L. Katzarkov, and V. Przyjalkowski, Birational geometry via moduli spaces.- O. Debarre, Curves of low degrees on projective varieties.- S. Kebekus, Uniruledness criteria and applications.- S. Kovács, The cone of curves of K3 surfaces revisited.- V. Lazi, Around and beyond the canonical class.- C. Liedtke, Algebraic surfaces in positive characteristic.- A. Varilly-Alvarado, Arithmetic of Del Pezzo surfaces.