Given a proper algebraic map f : X--> Y, the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber provides powerful tools to the study of its topology. This endows the cohomology of X with an extra structure, known as the perverse filtration, which measures the singularities of the map f. In recent years, the decomposition theorem and the induced perverse filtration have been found to share surprising connections to other branches of mathematics; these include non-abelian Hodge theory (the P=W conjecture), enumerative geometry (Donaldson-Thomas and BPS invariants), planar singularities (DAHA, knot invariants), and hyper-Kähler geometries (Hodge modules, motivic techniques). In this lecture series, I will discuss some of these developments. If time permits, open questions will be presented and discussed.