In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the
interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real
line). Most of the applications are based on an exten­ sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and
includes: creation of the theory of non-classical orthogonal polynomials with re­ spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of
incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the
asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational
approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully
develop the classical theory of logarithmic potentials.