Perverse sheaves and their associated invariants appear naturally in the study of certain abelian fibrations. A striking example is the P=W conjecture by de Cataldo, Hausel, and Migliorini connecting the topology of Hitchin systems and Hodge theory of character varieties via non-abelian Hodge theory. I will discuss recent progress on the P=W conjecture including a proof for the case of genus 2 and arbitrary rank, as well as connections to enumerative geometry for K3 surfaces and topology of hyperkaehler manifolds. Based on joint work with Qizheng Yin, and joint work in progress with Mark de Cataldo and Davesh Maulik.

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