The moduli spaces of semistable Higgs bundles on a curve have been inspiring lots of researchers across various fields, including Geometry, Topology, Representation Theory, Number Theory, and Mathematical Physics. They have been indeed one of the wealthiest sources, contributing areas of research such as non-abelian Hodge theory, the P=W conjecture, the SYZ style topological Mirror symmetry for the Hitchin fibrations, the Langlands programme and its geometric one, and more. In this talk, I would like to discuss intriguing analogues of them on a surface, strongly motivated by the Kapustin-Witten and Vafa-Witten theories from topologically twisted versions of N=4 super Yang-Mills one.