Teichmüller curves are totally geodesic complex curves inside the moduli space of Riemann surfaces. Under a Prym-Torelli map, one can see them as curves in some Hilbert modular variety, which parametrises abelian varieties with real multiplication. We give an overview of how to cut out these curves as the vanishing locus of Hilbert modular forms and the invariants that can be computed from this viewpoint.