This reference book provides the main definitions, theorems and techniques in the theory of Birkhoff interpolation by polynomials. The book begins with an article by G. G. Lorentz that discusses some of the important developments in approximation and interpolation in the last twenty years. It presents all the basic material known at the present time in a unified manner. Topics discussed include: applications of Birkhoff interpolation to approximation theory, quadrature formulas, and Chebyshev systems; lacunary interpolation at special knots; and an introduction to the theory of Birkhoff interpolation by splines.