The Voronoi formula is an analog of Poisson summation formula for automorphic forms. Previously, one Voronoi formula was known for Maass forms on GL(n). It has great application in analytic number theory of automorphic forms and their L-functions. We discover new types of Voronoi formulae for automorphic forms on GL(n) for n>=4. There are [n/2] different Voronoi formulae on GL(n), which are summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums of dimension k=1,2,…,[n/2].