Abstract. Cortical spreading depression (CSD) is a slowly propagating wave of ionic and metabolic disturbances in cortical brain tissue. In addition to massive cellular depolarization, CSD involves significant changes in tissue perfusion and metabolism. CSD has been linked to migraine with aura, which affects about 20% of the people who suffer from migraine. The triggers for this disease are mainly undiagnosed. To devise rational treatments of migraine with aura, we need to learn much more about the brain and about CSD. CSD was discovered almost 70 years ago by A.A.P. Leao, a Brazilian physiologist during his PhD research on epilepsy at the Harvard Medical School. CSD is characterized by nonlinear chemical waves that propagate at very slow speeds, on the order of mm/min, in the cortex of different brain structures in various experimental animals, and occurs in humans. CSD waves generate massive changes in extracellular ion concentrations. In this talk, I will review some of the characteristics of CSD wave propagation and describe some of the mechanisms that are believed to be important for CSD. We develop a new mathematical model for CSD where the sodium-potassium ATPase, responsible for cellular polarization and recovery from CSD, operates at a rate that is dependent on local oxygen concentration. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree. Our model replicates the qualitative and quantitative behavior of CSD found in experimental studies and elucidates the effect of oxygen deprivation on CSD recovery. Our key findings are that during CSD, the metabolic activity of the cortex exceeds the physiological limits placed on oxygen delivery and changes in perfusion alter the intensity and duration of the event. The combination of experimentation and modeling should accelerate our understanding of how these mechanisms conspire to form CSD.