This text has been revised due to renewed interest in the topic because of its use as a tool in numerical analysis and to computer scientists working in approximation theory. The following new material is included: a chapter on algebraic and number-theoretic properties of the Chebyshev polynomials; a chapter describing the historical role of the Chebyshev polynomials in the design of linkages, properties of the graphs of the Chebyshev polynomials, and multivariate generalizations of the polynomials.