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In the most general setting, optimization seeks to find a best option out of a set of feasible options. Very often, options consist of a complex combination of atomic decisions. If some of these decisions are, e.g., yes/no-decisions or decisions on integral quantities, then the optimization problem is discrete (as opposed to continuous). For example, your favorite navigation system solves discrete optimization problems whenever you ask for a route, since you can only go this way or that way and not half this way and half that way at the same time. In this lecture, after a brief introduction, many acadamic and real-world examples for discrete optimization are presented that show how far one can get with this tool today.