We survey results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface. After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus g, then its cop number is at most g + 3. While Schroeder's bound is known to hold for planar and toroidal graphs, the case for graphs with higher genus

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... emains open. We consider the capture time of graphs on surfaces and examine results for embeddings of graphs on non-orientable surfaces. We present a conjecture by the second author, and in addition, we survey results for the lazy cop number, directed graphs, and Zombies and Survivors.