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There are a vast number of books on mathematics, so why should you read this one? That is, of course, the million dollar question, which I will now try to answer. I should first say that I doubt that anything I say in this book has not been said many times before, and often in much greater detail. Indeed, in sketching historical and philosophical ideas relating to mathematics, I am very much aware that a huge amount could, and indeed has been, said already. If you find yourself interested in these topics, there are many excellent books on the history of mathematics, and also a host of discussions of a more philosophical nature on what mathematics is. The main aim of this book is to provide a straightforward introduction to topics such as groups and finite fields to engineers and computer scientists who find themselves working with—and possibly implementing—some of the strange types of arithmetic involved. These arithmetics are fundamental to much modern information and communications technology. Why is this book necessary? After all, there are plenty of excellent books that provide a grounding in mathematics for engineers and computer scientists. However, existing books cover what is commonly known as ‘traditional’ applied mathematics, that is very different to the topics that I cover in this book. While applied mathematics in the traditional sense is still of huge importance, modern digital systems increasingly build on very different types of mathematics, and these very different types of mathematics form the focus of this book. I have attempted to make the book accessible to as large an audience as possible. I will provide some proofs (it wouldn’t be a mathematics book without proofs) but if you like you can skip them, although I think it would be a shame to avoid them altogether since they are such an integral part of mathematics. More importantly, in all the main chapters after the introductory material I provide case studies of how the mathematics I introduce underlies key modern digital technologies, in particular focussing on those relating to secure and reliable communications. A somewhat subsidiary aim of the book is to introduce the general reader to some of the fascinating mathematics underlying the modern digital and information-centric world. In doing so, I am hoping to give non-mathematicians a taste of what mathematics is really about, at least as mathematicians see it. I should say that I first wanted to write this book at least ten years ago, but finding the time has been the main obstacle. Perhaps the fact that I still wanted to write this book after all this time means that it was a good idea! Anyway, I leave that for you to decide.