In this chapter, we discuss two loosely related topics: algorithms for com­ puting the special functions of mathematical physics (Bessel functions, orthogonal polynomials, etc.) and efficient methods of quadrature based on orthogonal functions. In most scientific computing, large libraries sup­ ply almost all of the subroutines relevant to these tasks and so relieve the individual from the tedium of writing code. In fact, there is usually little need to know very much about how these subroutines work in detail. However, a rough idea of the methods used is useful; this is what we hope to impart in this chapter.