A unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial differential and integral equations. Topics include: preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches including multigrid algorithms, hierarchical low-rank matrix decompositions, and low and high frequency Fast Multipole Methods. Example applications include: aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.