This article surveys recent advancements in strategy designs for persistent robotic surveillance tasks, with a focus on stochastic approaches. The problem describes how mobile robots stochastically patrol a graph in an efficient way, where the efficiency is defined with respect to relevant underlying performance metrics. We start by reviewing the basics of Markov chains, which are the primary motion models for stochastic robotic surveillance. We then discuss the two main criteria regarding the

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... peed and unpredictability of surveillance strategies. The central objects that appear throughout the treatment are the hitting times of Markov chains, their distributions, and their expectations. We formulate various optimization problems based on the relevant metrics in different scenarios and establish their respective properties. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.