A knot is simply a tangled loop in ordinary three-dimensional space, such as often causes us frustration in everyday life. Knots are also the subject of a rather rich mathematical theory. In the last three decades, it has unexpectedly turned out that rather deep aspects of the theory of knots are best understood in the context of 20th and 21st century developments in quantum physics. In this talk, Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, attempts to explain what quantum theory has to do with knots. He starts at the beginning in a talk that should be accessible to anyone who has ever puzzled over a particularly vexing mess of string.