Abstract. Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

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In 1945, W. Taylor discovered the barycentric formula for evaluating the interpolating polynomial. In 1984, W. Werner has given first consequences of the fact that the formula usually is a rational interpolant. We review some recent advances in the use of the formula for rational interpolation.

Abstract. In 1988 the second author presented experimentally well-conditioned linear rational functions for global interpolation. We give here arrays of nodes for which one of these interpolants converges exponentially for analytic functions

Well-conditioned, stable and infinitely smooth interpolation in arbitrary nodes is by no means a trivial task, even in the univariate setting considered here; already the most important case, equispaced points, is not obvious. Certain approaches have nevertheless experienced significant developments in the last decades. In this paper we review one of them, linear barycentric rational interpolation, as well as some of its applications.