Neuroscience is the area of mathematical biology in which mathematics has had perhaps the greatest impact. There now exist detailed models for electrical activity in neurons and in the brain; these models have motivated the development of new mathematical methods. In these lectures, I will describe some of these methods, along with biological issues that motivated them. In particular, I will discuss the Hodgkin-Huxley equations and tools from the geometric theories of dynamical systems and singular perturbations.