Let G be a graph in which each vertex has been coloured using one of k colours, say C1, c2, …, ck. If a graph H in G has ni vertices coloured ci i = 1, 2, ... , k, and |ni –nj| 5, there exists at least one 5-cycle decomposition of Kv which cannot be equitably 2-coloured. In addition, we completely settle the existence question for equitably 2-colourable m-cycle decompositions of Kv - F and for equitably 3-colourable m-cycle decompositions of Kv and Kv - F, where m Є { 4, 5, 6}. We also provide upper bounds on admissible values of v for existence of equitably (m - 1)-colourable m-cycle decompositions of Kv and Kv - F. In Chapter 3, we partially generalise our results on equitably 2-colourable even-length cycle decompositions of Kv - F. Except for the case where v(v - 2)/2 is an odd multiple of m and v = m = 4 (mod8), we show that if the obvious necessary conditions are satisfied then there exists an equitably 2-coloured m-cycle decomposition of Kv - F for even m. In Chapter 4, we completely settle the existence question for equitably 2-colourable 3- and 5-cycle decompositions of Kp(n), and for equitably 2-colourable 4- and 6-cycle decompositions of Kn1,n2, ... ,np,· In addition, we completely settle the existence question for equitably 3-colourable m-cycle decompositions of Kp(n), for m Є {3, 4, 5}. In Chapter 5, we completely settle the existence question for equitably 2- and 3-colourable 3-cube decompositions of Kv, Kv - F and Kx,y· We also consider other types of coloured graph decompositions. Suppose that the vertices of Kv, have been coloured with at most two colours. Let C1 C2, . . .Cm denote the colouring of the m-cycle (x1,x2, . . . ,xm) which assigns the colour Ci to the vertex xi for i = 1, 2, . . . , m, where Ci Є {black, white}. We let T be the set of all possible such colourings and we let S C T. If ɧ is an m-cycle system such that the colouring type of every m-cycle in ɧ is in S, and every colouring type in S is represented in ɧ, then we say that ɧ has proper colouring Type S. In Chapter 6, we completely settle the existence question for 4-cycle decompositions with proper colouring Type S for all possible S. In Chapter 7, we completely settle the existence question for 5-cycle decompositions with proper colouring Type S for all S when |S| = 1, and for many |S| when |S| = 2. For the remaining S, where |S| = 2, we determine some necessary conditions for existence of such decompositions.