The goal of this talk is to bring to light some recently discovered connections between problems about graph colourings and problems about the approximation of real numbers by rationals. The connections arise as a result of the fact that many statements about the quality of approximation of real numbers can be phrased as problems about the orbits of points in certain spaces (e.g. compact metric spaces) under the action of particular groups. Once these group actions are identified, there is a correspondence between questions about the approximations and questions about the Cayley graph of the given group. Information on either side of this correspondence gives information about the other. Several authors have used this machinery to transfer information from Diophantine approximation to give upper bounds for the chromatic number of Cayley graphs. Our main result shows that interesting information about Diophantine approximation can also be obtained by going in the other direction.We show how lower bounds for the chromatic number of certain Cayley graphs can be used to give a new proof of the p-adic Littlewood Conjecture for quadratic irrationals.