In our attempts to understand the circadian system, we unavoidably rely on abstractions. Instead of describing the behavior of the circadian system in all its complexity, we try to derive basic features from which we form a global concept on how the system works. Such a basic concept is a model of reality. The author discusses why it is advantageous or even necessary to transform conceptual models into mathematical formulations. As examples to demonstrate those advantages, the author reviews 4 types of mathematical models: negative feedback models thought to operate within pacemaker cells, models on coupling between pacemaker cells to generate pacemaker output, oscillator models describing the behavior of the composite circadian pacemaker, and models describing how the circadian pacemaker influences behavior.