Number theory is the branch of mathematics that is primarily concerned with the counting numbers. Of particular importance are the prime numbers, the building blocks of our number system. The subject is an old one, dating back over two millennia to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no one has ever been able to resolve them. But number theory has also recently become of great practical importance--in the area of cryptography, where the security of your credit card, and indeed of the nation's defense, depends on a result concerning prime numbers that dates back to the eighteenth century. Recent years have witnessed other spectacular developments, such as Andrew Wiles's proof of Fermat's last theorem (unproved for over 250 years) and some exciting work on prime numbers. In this Very Short Introduction Robin Wilson introduces the main areas of classical number theory, both ancient and modern. Drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler, and Gauss, he situates some of the most interesting and creative problems in the area in their historical context.