  This report is an introduction to mathematical map colouring and the problems
posed by Heawood in his paper of 1890. There will be a brief discussion of the
Map Colour Theorem; then we will move towards investigating empire maps in the
plane and the recent contributions by Wessel. Finally we will conclude with a
discussion of all known results for empire maps on higher genus surfaces and
prove Heawoods Empire Conjecture in a previously unknown case.
