One of the most popular domains in the theory of approximation of functions by means of rational functions is the theory of the Pade approximations. There exist many books and papers which consider this type of approximations. We want only to mention the excellent monograph in two volumes of Baker and Graves-Morris (1981). Here we want to consider some problems connected with the convergence of the Pade approximants, which are not entirely included in that monograph. These results are due to A.A. Gonchar and the group of mathematicians headed by him.

In section 12.1 we give the definition and some promerties of Padé approximants. In section 12.2 we have direct results for the convergence of Pade approximants - the classical theorem of Montessus de Ballore and one of its generalizations, which is due to A.A. Gonchar (1975a). In section 12.3 we give one converse theorem for the convergence of Pade approximants with fixed degree of denominator (the rows of the Pade-table) which is due to Gonchar (unpublished). In section 12.4 we give one more converse theorem of Gonchar connected with the diagonal of the Pade-table. In the notes to the chapter we give some more information about these problems.