We will discuss cohomological structures of the moduli space of Higgs bundles associated with a curve. On one hand, the Higgs moduli space can be viewed as an enhancement of the moduli of vector bundles. They share a lot of common features in geometry and topology. On the other hand, the Higgs moduli space admits the structure of an algebraically integrable system, called the Hitchin fibration, whose fibers are global analogs of affine Springer fibers. Hence the cohomological study of Higgs moduli spaces have rich connections with orbital integrals (Ngô), mirrory symmetry (Hausel-Thaddeus), and non-abelian Hodge theory (de Cataldo-Hausel-Migliorini). We will discuss these connections and some recent progress. Based on joint work with Mark de Cataldo and Davesh Maulik.