A colouring of a graph isnonrepetitiveif for every path of even order, thesequence of colours on the first half of the path is different from the sequence of colours onthe second half. We show that planar graphs have nonrepetitive colourings with a boundednumber of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan(2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding afixed minor, and graphs excluding a fixed topological minor.