Migraine with aura affects about 20% of the people who suffer from migraine (a severe headache affecting people around the world). The triggers for this disease are mainly undiagnosed and treatment is generally ad hoc from patient to patient. Migraine with aura has been linked to waves of cortical spreading depression (CSD) in the visual cortex of the brain. To devise rational treatments of migraine with aura, we need to learn much more about the brain and about CSD. We can learn a lot about the brain by studying extreme phenomena, such as CSD. CSD was discovered over 65 years ago by A.A.P. Leão, 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. To date, we do not have a good explanation of how CSD occurs, although a number of mechanisms have been hypothesized to be important for CSD wave propagation. In this talk, I will review some of the characteristics of migraine with aura and CSD wave propagation, and describe some of the mechanisms that are believed to be important for CSD, including ion diffusion, membrane ionic currents, osmotic effects, the spatial buffer mechanism, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. The emphasis will be on developing dynamical system models and continuum models of CSD, consisting of coupled nonlinear diffusion equations. (Work done in collaboration with H. Huang, York University, Toronto, Canada and W. Yao, Fudan University, Shanghai, China.)