The lambda-calculus lies at the very foundations of computerscience. Besides its historical role in computability theory it hashad significant influence on programming language design andimplementation, denotational semantics, and domain theory. The bookemphasises the proof theory for the type-free lambda-calculus. Thefirst six chapters concern this calculus and cover the basic theory,reduction, models, computability, and the relationship between thelambda-calculus and combinatory logic. Chapter 7 presents a varietyof typed calculi; first the simply typed lambda-calculus, thenMilner-style polymorphism and, finally, the polymorphiclambda-calculus. Chapter 8 concerns two variants of the type-freelambda-calculus that have appeared in the research literature: thelazy lambda-calculus, and the lambda sigma-calculus. The finalchapter contains references and a guide to further reading. There areexercises throughout. In contrast to earlier books on these topics,which were written by logicians, this book is written from a computerscience perspective and emphasises the practical relevance of many ofthe key theoretical ideas. The book is intended as a course text forfinal year undergraduates or first year graduate students in computerscience. Research students should find it a useful introduction tomore specialist literature.