We study the PBW filtration on the highest weight representations V(λ) of SLn+1. This filtration is induced by the standard degree filtration on U(n_). We give a description of the associated graded S(n_)-module gr V(λ) in terms of generators and relations. We also construct a basis of gr V(λ). As an application we derive a graded combinatorial character formula for V(λ), and we obtain a new class of bases of the modules V(λ) conjectured by Vinberg in 2005.