In the first part of my talk, I will introduce the P=W conjecture by de Cataldo, Hausel, and Migliorini (2010), predicting that the perverse filtration associated with the Hitchin system is identified with the weight filtration associated with the corresponding character variety, via non-abelian Hodge theory. This part can be viewed as a review of Davesh Maulik’s colloquium talk in November 9. In the second part, I will focus on a proof of the conjecture in recent joint work with Davesh Maulik. In particular, we will discuss the tools we used in algebraic geometry (critical loci and vanishing cycles, support theorem) and representation theory (global Springer theory) which lead to our solution to the conjecture.