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