The Mumford-Tate conjecture serves as a bridge between the analytic world of Hodge theory and the arithmetic world of Galois representations. While the conjecture is difficult even for abelian varieties, hyper-Kähler varieties offer a promising testing ground due to their similarity to K3 surfaces. In this talk, I will introduce the Mumford-Tate conjecture and the geometry of hyper-Kähler varieties. I will then present recent results (joint with Zhichao Tang) showing that the conjecture holds for the semisimplified Galois representations of even weights, provided the second Betti number is at least 4. This generalizes previous results known only for specific deformation types. If time permits, I will also sketch an application of this result to the purity of weights for monodromy filtration over local fields.