  This paper is a survey on the role of Higgs bundle theory in the study of
higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact
surface can be identified with a certain connected component of the moduli
space of representations of the fundamental group of the surface into
$\mathrm{PSL}(2,{\mathbb{R}})$. Higher Teichm\"uller spaces correspond to
special components of the moduli space of representations when one replaces
$\mathrm{PSL}(2,{\mathbb{R}})$ by a real non-compact semisimple Lie group of
higher rank. Examples of these spaces are provided by the Hitchin components
for split real groups, and maximal Toledo invariant components for groups of
Hermitian type. More recently, the existence of such components has been proved
for $\mathrm{SO}(p,q)$, in agreement with the conjecture of Guichard and
Wienhard relating the existence of higher Teichm\"uller spaces to a certain
notion of positivity on a Lie group that they have introduced. We review these
three different situations, and end up explaining briefly the conjectural
general picture from the point of view of Higgs bundle theory.
