The study of colour-biased structures in graphs concerns the following problem: given graphs H and G , what is the largest t such that, in any r-colouring of the edges of G, there is always a copy of H in G having at least t edges of the same colour?  In this talk, we present a generalisation of a recent result of Balogh, Csaba, Jing and Pluhár (2020) establishing the minimum degree threshold that ensures a 2-coloured graph G contains a Hamilton cycle H of a specified colour bias. We obtain the corresponding tight threshold for r-coloured graphs.  This is joint work with Andrea Freschi, Joseph Hyde, and Andrew Treglown (Birmingham).