Moduli spaces are geometric versions of parameter spaces. That is, they are geometric spaces which parametrise something --- each point represents one of the objects being parametrised, such as the solution of a particular equation, or a geometric structure on some other object. In the language of physics, a moduli space is a model of the degrees of freedom of some system of equations.

Moduli spaces of holomorphic vectors bundles over Riemann surfaces have been studied for more than 50 years, specially since the fundamental work of Mumford on Geometric Invariant Theory, and Narasimhan and Seshadri. In the last three decades, very much through the work of Atiyah--Bott, Donaldson, Hitchin and others, there has been a lot of work in the study of the relations between moduli spaces of vector bundles, principal bundles and similar objects like Higgs bundles ---also on higher dimensional Kaehler manifolds---, with other moduli spaces, like the moduli spaces of solutions of non-linear partial differential equations emerging from gauge theory in Particle Physics, and moduli spaces of representations of the fundamental group of a Kaehler manifolds in reductive Lie groups. The study of these relations is the content of the Hitchin--Kobayashi correspondences and non-abelian Hodge theory, much studied by the members of the ICMAT team, who have made important contributions also in the study of the topology and geometry of these moduli spaces.