The structure of the laws in physics is largely based on symmetries. This book is on Lie algebras, the mathematics of symmetry. It has grown from lectures for undergraduates in theoretical and mathematical physics and gives a thorough mathematical treatment of finite dimensional Lie algebras and Kac-Moody algebras. Concepts such as Cartan matrix, root system, Serre's construction are carefully introduced. Although the book can be read by an undergraduate with only an elementary knowledge of linear algebra, the book will also be of use to the experienced researcher. Experience has shown that students who followed the lectures are well-prepared to take on research in the realms of string-theory, conformal field-theory and integrable systems. The new series ``Studies in Mathematical Physics'' aims at discussing recent developments in physics offering sound mathematics and a high didactical quality. The emphasis lies on techniques, ideas and methods that are fundamental, interesting and innovating in both mathematics and physics, herewith creating a link between the two disciplines.