The weak Lefschetz property (WLP) plays an important role both in algebra and geometry. In [3], Mezzetti, Miró-Roig and Ottaviani found that the failure of the WLP for some particular Artinian ideals in K[x0, ... , xn] is related with the existence of projections of the Veronese variety satisfying one Laplace equation. This relation gives rise to the denition of Togliatti system. In this note, we state some recent results on this topic. In particular, we expose the classication of minimal smooth Togliatti systems generated by 2n +3 monomials of degree d ≥ 10 obtained in [8].