Neural activity in the brain is accompanied by changes in cerebral blood flow (CBF) and blood oxygenation that are detectable with functional magnetic resonance imaging (fMRI) techniques. In this paper, recent mathematical models of this hemodynamic response are reviewed and integrated. Models are described for: (1) the blood oxygenation level dependent (BOLD) signal as a function of changes in cerebral oxygen extraction fraction (E) and cerebral blood volume (CBV); (2) the balloon model, proposed to describe the transient dynamics of CBV and deoxy-hemoglobin (Hb) and how they affect the BOLD signal; (3) neurovascular coupling, relating the responses in CBF and cerebral metabolic rate of oxygen (CMRO(2)) to the neural activity response; and (4) a simple model for the temporal nonlinearity of the neural response itself. These models are integrated into a mathematical framework describing the steps linking a stimulus to the measured BOLD and CBF responses. Experimental results examining transient features of the BOLD response (post-stimulus undershoot and initial dip), nonlinearities of the hemodynamic response, and the role of the physiologic baseline state in altering the BOLD signal are discussed in the context of the proposed models. Quantitative modeling of the hemodynamic response, when combined with experimental data measuring both the BOLD and CBF responses, makes possible a more specific and quantitative assessment of brain physiology than is possible with standard BOLD imaging alone. This approach has the potential to enhance numerous studies of brain function in development, health, and disease.