Polynomials are very simple mathematical functions, which have the flexibility to represent very general nonlinear relationships. Approximation of more complicated functions by polynomials is a basic building block for many numerical techniques. This article considers two distinct but related applications. The first is polynomial regression in which polynomials are used to model a nonlinear relationship between a response variable and an explanatory variable. The advantages of using orthogonal polynomials as predictor variables are illustrated using a data set on the height and age of preadult girls. The second problem is that of approximating a difficult to evaluate function, such as a density or a distribution function, with the aim of fast evaluation on a computer. The use of Chebyshev polynomials is illustrated for the purpose of obtaining a uniformly accurate approximation to a function over a finite interval.