In this survey, we review some conjectures on the cohomology of the moduli space of Higgs bundles on a curve and explain joint work with O. Garcia-Prada and A. Schmitt, “On the motives of moduli of chains and Higgs bundles” and “The $y$-genus of the moduli space of $\textrm {PGL}_n$-Higgs bundles on a curve (for degree coprime to $n$)”, resulting in a recursive algorithm to determine the cohomology of moduli spaces of semi-stable Higgs bundles on a curve (in the coprime situation). This method allows us to confirm a conjecture of T. Hausel who predicted a formula for the $y$-genus of the moduli space.