We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g, where f(n) is the best approximation factor achievable in polynomial time on arbitrary n-vertex graphs. In particular, the O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation algorithm for genus-g graphs. Our result improves the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g; ours is the first approximation algorithm for graphs with bounded non-orientable genus. Moreover, using recent progress on approximating the genus of a graph, our O(log(g) / loglog(g))-approximation can be implemented even without an embedding when the input graph has bounded degree. In contrast, the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a genus-g embedding as part of the input. Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on graphs of genus g, with running time 2^O(g)*n^O(1).