Electronic structure calculations form the basis of material science, quantum chemistry, and molecular dynamics. Due to the high dimension of the underlying equations together with strict requirement on the accuracy of the solution, electronic structure calculations pose a unique challenge to mathematical analysis and numerical methods. In this seminar we will discuss fundamental models for the description of electrons and how they can be solved numerically. Topics include the discretization of Schrödinger's equation and the Kohn-Sham equations, the iterative solution of resulting linear and nonlinear eigenvalue problems, and the application of the methods to atoms, small molecules, and crystals. No prior knowledge of physics is necessary to attend this seminar.