This seminar-based course will focus on the use of popular iterative methods for the solution of general linear systems of equations. Using the Stanford University Technical Report NA-91-05 by Freund, Golub, and Nachtigal as our guide, we will survey Lanczos-based Krylov subspace methods for the solution of symmetric and nonsymmetric systems. The first 7 weeks of the course will comprise a general review of basic iterative methods (Jacobi, Gauss-Seidel, SOR, etc.,) and an introduction to well-known Krylov methods (Conjugate Gradient, Preconditioned Conjugate Gradient, Lanczos, Arnoldi, GMRES, etc.,) in order to adequately prepare students for a more detailed study of more recent methods such as CGS, Bi-CGSTAB, and QMR. Appropriate survey and research papers (plus references) will be distributed to all registered students during the semester.