  Cops and Robbers is a well-studied pursuit-evasion game in which a set of
cops seeks to catch a robber in a graph G, where cops and robber move along
edges of G. The cop number of G is the minimum number of cops that is
sufficient to catch the robber. Every planar graph has cop number at most
three, and there are planar graphs for which three cops are necessary [Aigner
and Fromme, DAM 1984]. We study the problem for beyond-planar graphs, that is,
graphs that can be drawn in the plane with few crossings. In particular, we
focus on 1-planar graphs, that is, graphs that can be drawn in the plane with
at most one crossing per edge. In contrast to planar graphs, we show that some
1-planar graphs have unbounded cop number. Meanwhile, for maximal 1-planar
graphs, we prove that three cops are always sufficient and sometimes necessary.
In addition, we characterize outer 1-planar graphs with respect to their cop
number.
