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We introduce a novel optimization problem to support the planning and routing of on-demand buses in an urban context. We call this problem the on-demand bus routing problem (ODBRP). Given are a fleet of buses with fixed capacity, a set of bus stations and travel times between them, and a set of transportation requests. Each transportation request consists of a set of potential departure and a set of potential arrival bus stations, as well as a time window, that is, an earliest departure time and a latest arrival time. The aim of the ODBRP is to (1) assign each passenger to a departure and arrival bus station and (2) develop a set of bus routes to fulfill each request in time while minimizing the total travel time of all users. We present the static version of the ODBRP, as well as a straightforward large neighborhood search heuristic to solve it. The performance of the heuristic is established by comparing it to an off-the-shelve heuristic solver (LocalSolver). We also use our heuristic to solve (slightly modified instances of) the well-known dial-a-ride problem. The results found by the heuristic for the on-demand bus system are compared to those of a simulated traditional public bus system with fixed lines and timetables. A thorough analysis of the comparison demonstrates that total user ride times can be significantly lower in an on-demand public bus system and shows that an on-demand bus system works best with a large number of small buses.