It is classically known that the Teichmüller space of a smooth, closed, oriented surface of genus g>1 is a ball of dimension 6g-6. Various Riemannian (and non-Riemannian) metrics with interesting geometric properties have been introduced on it. One of these is the Weil-Petersson metric defined in terms of holomorphic quadratic differentials on the surface, which turns out to be Kähler. In this talk, we will see how such a metric has a natural extension on the Hitchin component for PSL(3,R), in which the Teichmüller space is naturally contained, and what some of its remarkable properties are. Throughout the exposition we will recall the basic definitions and the geometric interpretations of Hitchin representations in this particular case. This is a joint work with Andrea Tamburelli.