  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.
