Prime numbers - numbers like 2,3,5,7 or 53, which have no divisors - have been a source of fascination since the ancient Greeks. Although they lie primarily in the realm of the pure mathematician, they occur in the natural world, and are vital to modern life, being crucial to the security of internet banking. The talk will look at the history of prime numbers, their occurrence in the real world, and what mathematicians have to say about them.