This talk will be an overview on Higgs bundles moduli spaces and on the geometry of the Hitchin system, focusing on its connections to mirror symmetry. I will focus on the existence of certain special subvarieties of these moduli spaces equipped with special sheaves, called branes. These are conjecturally exchanged under mirror symmetry, thus being so-called dual branes. One type of such branes is constructed from hyperholomorphic subvarieties and their conjectural dual arise from Lagrangian subvarieties. Even though this phenomenon has been under intensive research in the last decade, it is still far from being well-understood, especially when those branes lie over the singular locus of the Hitchin system. I will report joint work with E. Franco, P. Gothen and A. Peón-Nieto where we explicitly described a pair of such dual branes and, time permitting, also on joint work in progress with E. Franco, R. Hanson and J. Horn where we give more examples of Lagrangian subvarieties on this singular locus and study their mirror symmetry dual.