Hitchin’s self-duality equation on Riemann surfaces was introduced almost four decades ago as a dimension reduction of the self-dual instanton equation in 4D. I will begin with an overview of the rich geometry of the moduli space of its solutions, its incarnations as moduli space of Higgs bundles and of representations of the fundamental group, as well as Hitchin fibration and spectral curve. I will then discuss a series of recent works on limiting behavior of solutions, motivated by questions about the asymptotic geometry of the moduli space.
