This book is a unique reference for anyone with a serious interest in mathematics. It is edited by Timothy Gowers, a recipient of the Fields Medal, and it includes entries written by several of the world's leading mathematicians. The entries have many goals, such as to introduce basic mathematical tools and vocabulary, to trace the development of modern mathematics, to explain essential terms and concepts, to describe the achievements of scores of famous mathematicians and to explore the impact of mathematics on other disciplines such as biology, finance and music. The book is divided into eight chapters. The introduction presents basic mathematical language and some fundamental mathematical definitions. The origins of modern mathematics are described in the second chapter. It starts with numbers, geometry and the development of abstract algebra and through algorithms and proofs in mathematics gets to the crisis in the foundations of mathematics. The next chapter is devoted to important mathematical concepts presented in alphabetical order. Thus, one can begin reading with an axiom of choice and end with the Zermelo-Fraenkel axioms. Then the book turns its attention to mathematical branches. From algebraic geometry to stochastic processes, brief descriptions of the main areas of mathematics are provided. Crucial theorems and fundamental open problems occupy the fifth chapter. Then the book presents some achievements of famous mathematicians, starting with Pythagoras and ending with Nicolas Bourbaki. After this, several examples of the influence of mathematics on other branches of science are presented. The companion ends with general thoughts on mathematics and mathematicians. The book contains some valuable surveys of the main branches of mathematics that are written in an accessible style. Hence, it is recommended both to students of mathematics and researchers seeking to understand areas outside their specialties.