How well can the concept of Padé approximant be generalized to the multivariate case?

What we know about multivariate Padé approximation has been developed in the last 25 years. In the next sections we compare and discuss many of these results. It will become clear that simple properties or requirements, such as the uniqueness of the Padé approximant and consequently its consistency property, can play a crucial role in the development of the multivariate theory. A separate section is devoted to a discussion of the convergence properties. At the end we include an extensive reference list on the topic.