In this monograph we present some results and problems in the modern theory of best approximation, i.e., in which the methods of functional analysis are applied in a consequent manner. This modern theory constitutes both a unified foundation for the classical theory of best approximation (which treats the problems with the methods of the theory of functions) and a powerful tool for obtaining new results. Within the general framework of normed linear spaces the problem of best approximation amounts to the minimization of a distance, which permits us to use geometric intuition (but rigorous analytic proofs), and the connections of the phenomena become clearer and the arguments simpler than those of the classical theory of best approximation in the various particular concrete function spaces. We hope that this has been proved convincingly enough in the monograph [168] (which was the first of this kind in the literature) and in the lecture notes [175], and will be proved again in the present monograph (see, for example, § 1, the remarks made after Theorem 1.8, or § 3, the remark to Theorem 3.5).