For a Hyperkähler variety which admits a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse $t$-structure. We will discuss the role played by this filtration in the study of the topology and geometry of Hyperkähler varieties. First, we will focus on the perverse filtration for the moduli of Higgs bundles with respect to the Hitchin fibration. We will discuss our recent proof of de Cataldo, Hausel, and Migliorini’s $P=W$ conjecture for parabolic Higgs bundles labelled by affine Dynkin diagrams. Then I will present a surprising connection between the perverse filtration for a projective Hyperkähler variety and the (pure) Hodge structure on itself. Based on joint work with Qizheng Yin and Zili Zhang.