The Hodge conjecture is a central problem in modern algebraic geometry. It is notoriously difficult to attack, and we still lack general evidence towards its validity. In my talk I will present a proof of the Hodge conjecture for all six-dimensional hyper-Kähler varieties of generalized Kummer type, i.e. those arising via deformation of Beauville's generalized Kummer varieties built from abelian surfaces. The result presented yields the first complete families of projective hyper-Kähler varieties of dimension larger than two for which the Hodge conjecture is verified. As I will explain, a key ingredient for the proof is the construction of a K3 surface naturally associated to a sixfold of generalized Kummer type.