Abstract. This article surveys and gives historical accounts to the algorithmic essentials of directed model checking, a promising bug-hunting technique to mitigate the state explosion problem. In the enumeration process, successor selection is prioritized. We discuss existing guidance and methods to automatically generate them by exploiting system abstractions. We extend the algorithms to feature partial-order reduction and show how liveness problems can be adapted by lifting the search space. For deterministic, finite domains we instantiate the algorithms to directed symbolic, external and distributed search. For real-time domains we discuss the adaption of the algorithms to timed automata and for probabilistic domains we show the application to counterexample generation. Last but not least, we explain how directed model checking helps to accelerate finding solutions to scheduling problems. 1