This volume contains the proceedings of the AMS Special Session on Polytopes and Discrete Geometry, held from April 21-22, 2018, at Northeastern University, Boston, Massachusetts. The papers showcase the breadth of discrete geometry through many new methods and results in a variety of topics. Also included are survey articles on some important areas of active research. This volume is aimed at researchers in discrete and convex geometry and researchers who work with abstract polytopes or string $C$-groups. It is also aimed at early career mathematicians, including graduate students and...