This thesis considers the existence of certain vertex and edge colourings of oriented graphs G, primarily oriented colourings, given the topological structure of G. The main measures of the topology of G we consider is the Euler genus g, maximum degree ∆, and degeneracy d. In doing so we present results by the author appearing in [18, 21], particularly as they pertain to the oriented chromatic number of a graph G, denoted χo(G). The first result is that for all k ≥ 2 and d ≥ log2(k), if G is a d-degenerate graph with χ2(G) ≤ k (2-dipath chromatic number), then χo(G) ≤ (33/10)kd^2 2^d, which greatly improves the prior bound of the form χo(G) ≤ 2k − 1 from MacGillivray, Raspaud, and Swartz [48] whenever log2(k) ≤ d ≪ k. Additionally we give constant factor asymptotic improvements on bounds for χo(G) in terms of maximum degree and degeneracy from Kostochka, Sopena, and Zhu [46] as well as Aravind and Subramanian [8]. Our final and largest contribution is to prove that the oriented chromatic number is at most g^{6400} for all graphs with Euler genus at most g, improving the prior asymptotic upper bound χo(G) ≤ 2O(g^{1/2+ε}) shown by Aravind and Subramanian [8].