Hermite-Padé approximation, multiple orthogonal polynomials, and multidimensional Toda equations

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.