This book, written in two volumes, is an introductory manual for Magma, a programming language designed for the investigation of algebraic, geometric and combinatorial structures, or "magmas." It presumes no knowledge of computer programming, and its examples are chosen to illustrate language and algorithmic features as simply as possible. Magma is a sophisticated tool for experimentation, education, and computer-aided proof, useful for both students and professional mathematicians. Volume I explains the language and user environment in detail, whereas Volume II deals with the major algebraic, geometrical and combinatorial structures implemented in the system.