During the first part of the talk I will give a gentle introduction to the theory of cohomological Hall algebras and their relevance in the study of the topology of moduli spaces, such as the Hilbert schemes of points on a smooth surface. The second part of the talk is devoted to the definition of the 2-dimensional cohomological Hall algebras of curves and surfaces. If time permits, I will discuss the construction of a categorification of these algebras. This is based on joint works with Olivier Schiffmann and Mauro Porta.