Patrolling and surveillance games both deal with a chasing-evading situation of an adversary trying to escape detection by either a mobile defender (patrolling) or a fixed defender (surveillance). Both kinds of games are played on graphs as abstract models of an infrastructure, and we review a variety of closed-form solutions for optimal patrolling in different classes of graph topologies. Applications include patrolling along lines (borders, pipelines, or similar), harbors (tree-structured graphs), and large geographic areas in general (planar graphs and maps). For surveillance and patrolling, we give hints on how to estimate the necessary resources, and how to include imperfectness and uncertainty, related to the detection capabilities, but also the chances of the adversary escaping the view of the patroller or surveillance. In complex terrain, we will discuss the use of simulation and empirical games (over real-valued and stochastic orders).