The purpose of this book and its sequel is to give a connected, unified exposition of Approximation Theory for functions of one real variable. Great care has been taken to provide reliable and good proofs, and to establish a logical selection of material, with emphasis on principal results. The history of the subject is not neglected. Methods of functional analysis are used when necessary, as are complex variable methods for real problems. Practical algorithms of approximation are included. Important old results are not missing, but at least half of the material has never yet appeared in books. The first book describes spaces of functions: Sobolev, Lipschitz, Besov rearrangement-invariant function spaces, interpolation of operators. Then we have Weierstrass and best approximation theorems, properties of polynomials and of splines: inequalities, interpolation (also Birkhoff), zero properties. Approximation by operators treats positive operators, projections, also those of minimal norm, saturation phenomena, a modern theory of Bernstein polynomials. Chapters on splines deal with the selection of knots, splines with equidistant, dyadic, fixed and free knots, with identification of approximation spaces, further with orthogonal projections onto splines, cardinal splines, shape preserving algorithms. With no equally comprehensive and up-to-date competitor in the available literature, these two volumes will be a welcome reference for a wide audience of mathematicians and engineers. They can be used as the basis for courses, or as a means to enter research in the subject.