This survey of the most important properties of the Chebyshev polynomials encompasses several areas of mathematical analysis: interpolation theory, orthogonal polynomials, approximation theory, numerical integration, numerical analysis, and ergodic theory. Originally published in 1974, the text was updated in 1990; this reprint of the second edition corrects various errors and features new material, including a chapter introducing elementary algebraic and number theoretic properties of Chebyshev polynomials.