In 1900, pure mathematics had the smug air of a finished product. We thought we knew what it was and we thought we knew how it was done. Then Bertrand Russell came along with an analysis that has the feeling of a childhood paradox, and blew the smugness away, perhaps forever. In this short talk I shall describe the pre-Russell situation, and go through his paradox in a way that you will find accessible. After that, I shall attempt an informal account of what happened next before ending with some of the work of Alan Turing. Dr Bob Lockhart is Director of Studies in Computing and Mathematics and Course Director of our Undergraduate Advanced Diploma in Data and Systems Analysis.