Padé approximants and efficient analytic continuation of a power series

This survey reflects the current state of the theory of Padé approximants, that is, best rational approximations of power series. The main focus is on the so-called inverse problems of this theory, in which one must make deductions about analytic continuation of a given power series on the basis of the known asymptotic behaviour of the poles of some sequence of Padé approximants of this series. Row and diagonal sequences are studied from this point of view. Gonchar's and Rakhmanov's fundamental results of inverse nature are presented along with results of the author.