Abstract: In this introductory talk I will try to give an idea of the different approaches to moduli spaces of Higgs bundles introduced by Hitchin and how the combination of these viewpoints gave rise to a series of fascinating questions and conjectures on the global geometry of these moduli spaces. We will then try to indicate some of the recent progress on these questions. As asked by the organizers, I will try to give an introductory overview, focussing mostly on various algebraic methods.