We review recent results on the connection between Hermite-Pad\'e approximation problem, multiple orthogonal polynomials, and multidimensional Toda equations in continuous and discrete time. In order to motivate interest in the subject we first present a pedagogical introduction to the classical, by now, relation between the Pad\'e approximation problem, orthogonal polynomials, and the Toda lattice equations. We describe also briefly generalization of the connection to the interpolation problems and to the non-commutative algebra level.

翻译：本文综述了埃尔米特-帕德逼近问题、多重正交多项式与连续及离散时间多维Toda方程之间关联的最新研究进展。为激发对该课题的兴趣，我们首先通过教学式阐述介绍了经典的帕德逼近问题、正交多项式与Toda晶格方程之间的相互关系。此外，本文还简要描述了该关联向插值问题及非交换代数层面的推广。