Chebysev polynomials are important in approximation theory and in the development of spectral methods for the solution of ordinary and partial differential equations in fluid dynamics and time dependent problems. This book gives an up-to-date exposition on the theoretical and practical roles of the Chebyshev polynomials. Theoretical discussions focus on definitions and properties as well as the key formulae for generating the polynomials and computing the expressions that involve them. Practical discussions center on applications that include polynomial approximation, rational approximation, integration, integral equations, and ordinary and partial differential equations, particularly the tau and spectral methods.