Part I: Approximating Functions, Derivatives and Integrals. 1. Spectral Approximation. 2. Algorithms For Periodic Functions. 3. Algorithms For Non-Periodic Functions. Part II: Approximating Solutions of PDEs. 4. Survey of Spectral Approximations. 5. Spectral Approximation on the Square. 6. Transformation Methods from Square to Non-Square Geometries. 7. Spectral Methods in Non-Square Geometries. 8. Spectral Element Methods. A: Miscellaneous Algorithms. B. Floating Point Arithmetic. C. Basic Linear Algebra Subroutines (BLAS). D. Linear Solvers. E. Data Structures.