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：本文综述了连续与离散时间下Hermite-Padé逼近问题、多重正交多项式与多维Toda方程之间关联的最新研究成果。为激发对该主题的兴趣，首先对经典的Padé逼近问题、正交多项式与Toda格方程之间的关联进行了教学性介绍，并简述了这一关联在插值问题及非交换代数层面的推广。