We survey the utility and function of mathematical and computational models in cognitive science by emphasizing their role as a reasoning aid for researchers. Unlike purely verbal theorizing, the rigor of computational models forces researchers to understand the implications of theoretical constructs. We review four types of models: (1) Descriptive models, which replace the raw data with parameter estimates but are not constrained by any prior theoretical assumptions. (2) Measurement models, which are also fitted to individuals or experimental conditions by unconstrained parameter estimation, but that embody strong structural assumptions that constrain the way in which dependent variables in an experiment are related. (3) Explanatory models, which go beyond measurement models by postulating not only the processes that operate in a single condition in an experiment but also explain how and why experimental conditions differ from each other. (4) Finally, cognitive architectures. which seek to explain a broad range of cognitive processes and activities at a higher level of (often symbolic) abstraction. For all types of model, we show how they can be related to the neural underpinnings of cognition.