This is a textbook on classical polynomial and rational approximation theory for the twenty-first century. It is aimed at advanced undergraduates and graduate students across all of applied mathematics.

The following are the distinctive features of the book:

The emphasis is on topics close to numerical algorithms.

Everything is illustrated with Chebfun.

Each chapter is a publishable Matlab M-file, available online.

There is bias towards theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges.

Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment.