This article widens the scope of the cell-transmission model (CTM) to the modeling of urban traffic, which is dominated by intersection dynamics. The CTM has originally been proposed for freeways, and it is not immediately applicable to urban traffic for the following reasons: Its intersection model is limited to three-legged topologies and ignores all stream line interactions apart from pure inflow capacity constraints of downstream merges, and, like every first order model, the CTM does not impose limits on vehicle accelerations and decelerations. However, the CTM has a number of important advantages that motivate an effort to carry it over to urban traffic instead of setting up an entirely new model: It is computationally efficient, requires only few, intuitive parameters, and, last but not least, it is well-understood by both researchers and practitioners. The realistic modeling of urban traffic with the CTM requires to specify the model for arbitrary intersection topologies without affecting its basic principles and to incorporate additional flow constraints that capture stream line interactions within intersections. The model's infinite accelerations are an intrinsic property of all first order models, and hence it appears more plausible to milden their negative effects by appropriate parametrization instead of attempting to move the CTM out of this model class. Finally, a framework is desirable that allows for incremental phenomenological modeling based on a basic specification of greatest generality. This article treats all of these issues in detail and demonstrates that the CTM is applicable to the modeling of complex urban intersections.