Since the second edition of this book, published in 1977, went out of print I have received what a less modest person might describe as fan-mail, lamenting the fact that a really accessible introduction to algebraic topology, through simplicial homology theory, was no longer available. Despite the lapse of years, and despite the multiplicity of excellent texts, nothing quite like this book has appeared to take its place. That is the reason why the book is being reprinted, newly and elegantly typeset, and with all the figures re-drawn, by Cambridge University Press. (I don't think TEX was even a gleam in the eye of that wonderful benefactor of mathematicians, Donald Knuth, when the first edition of the book was beautifully typed on a double keyboard mechanical typewriter by the then Miss Ann Garfield, now Mrs Ann Newstead.)

Of course, homology theory has advanced in the interim – not just the theory, but most importantly the multiplicity of applications and interactions with other areas of mathematics and other disciplines. Shape description, robotics, knot theory (Khovanov homology), algebraic geometry and theoretical physics – to name just a few areas – use topological ideas and in particular homology and cohomology. A quick search with an Internet search engine will turn up many references to applications. Since this book is intended for beginners, there is no pretence of being able to cover the recent developments and I have confined my ‘updating’ to correcting obvious errors, replacing references to other textbooks with more accessible modern ones, including additional references to the research literature and adding some comments where it seemed appropriate.