The relationship between ventilation and oxygen saturation during hypoxia is usually assumed to be approximately linear, but when the ventilatory response to hypoxia, expressed as ventilation/oxygen saturation, was measured for 10 subjects, in 8 of the subjects it was found to be greater when measured using step change hypoxia than when using transient hypoxia. Using Fourier techniques, the ventilation/saturation relationship was shown not to be linear. Use of a detailed physiological model failed to reveal a cause for the difference between the response to step and transient hypoxia, and rate of fall of oxygen saturation was shown experimentally and theoretically not to be an important factor in the difference. In the absence of a clear quantitative explanation for the discrepancy, a mathematical model was developed to describe the dynamic ventilatory response. The model was built by adding terms of increasing complexity to a simple linear differential equation. The simplest model which adequately described the responses of all the subjects consisted of two linear differential equations (1 and 2) in parallel, the input of both being the fall in oxygen saturation, the sum of the outputs giving the rise in ventilation. Equation 1 had a fast time constant (&60 3 sec), and Equation 2, a slow time constant. Non-linear terms included were a 'saturating effect', similar to that described by the Michaelis-Menten formulation, which reduced gain 2 as oxygen saturation fell; and 'inhibition' or 'potentiation' of gain 1 as the output of Equation 2 increased. This model produced statistically better fits to the data than any of the simpler models tried. As well as providing a more precise description of the hypoxic ventilatory response, the model suggested further experiments that might elucidate the physiological mechanisms occurring during hypoxia. Using a further model, an appendix discusses flaws in the widely-used technique of attempting to control arterial carbon dioxide tension by using end-tidal carbon dioxide pressure as a controlled variable.