Moduli spaces of Higgs bundles were introduced by Hitchin in 1987 and are now ubiquitous in geometry. They are closely related to other fundamental spaces such as character varieties, and they carry an incredibly rich structure, including a hyperkähler metric (a Riemannian metric compatible with three complex structures). In this talk I will give a gentle introduction to all of these objects, and describe some recent progress towards describing the metric near the singular locus of the Hitchin fibration.