This volume is a compilation of the research produced by the International Group for the Psychology of Mathematics Education (PME) since its creation, 30 years ago. It has been written to become an essential reference for Mathematics Education research in the coming years.The chapters offer summaries and synthesis of the research produced by the PME Group, presented to let the readers grasp the evolution of paradigms, questions, methodologies and most relevant research results during the last 30 years. They also include extensive lists of references. Beyond this, the chapters raise the main current research questions and suggest directions for future research.The handbook is divided into five sections devoted to the main research domains of interest to the PME Group. The first three sections summarize cognitively oriented research on learning and teaching specific content areas, transversal areas, and based on technology rich environments. The fourth section is devoted to the research on social, affective, cultural and cognitive aspects of Mathematics Education. Finally, the fifth section includes two chapters summarizing the PME research on teacher training and professional life of mathematics teachers.The volume is the result of the effort of 30 authors and 26 reviewers. Most of them are recognized leading PME researchers with great expertise on the topic of their chapter.This handbook shall be of interest to both experienced researchers and doctoral students needing detailed synthesis of the advances and future directions of research in Mathematics Education, and also to mathematics teacher trainers who need to have a comprehensive reference as background for their courses on Mathematics Education.