This special topics course will cover numerical methods for solving the large sparse systems of linear algebraic equations that are ubiquitous in scientific computing today. Direct methods such as Gaussian elimination are impractical because of excessive work and storage requirements. Instead, iterative methods are used. The course will emphasize the theory behind Krylov space methods and their analysis. Interesting open problems, such as the derivation and analysis of near-optimal methods for nonsymmetric linear systems, will be discussed. Preconditioners will be covered, with special attention to multigrid methods.