We study the minimal free resolution of the Veronese modules, S n,d,k = ⊕ i≥0 S k+id , where S = K[x 1 , . . . , x n ], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that S n,d,k is Cohen-Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d(n − 1) − n. We prove combinatorially that the resolution of S 2,d,k is pure. We show that HS(S n,d,k ; z) =As an application, we calculate the complete Betti diagrams of the Veronese rings K[x, y, z] (d) , for d = 4, 5, and K[x, y, z, u] (3) .Given a graded ring S = ⊕ i≥0 S i , the Veronese subring S (d) is defined as ⊕ i≥0 S id and the Veronese modules S n,d,k , which are modules over the Veronese subring, are ⊕ i≥0 S k+id . In this paper, we set S = K[x 1 , . . . , x n ], where K is a field, and we deal with the syzygies of the Veronese modules. Since S (d) can be presented as R/I, where R is a polynomial ring and I a binomial ideal, in the following, we will consider S n,d,k as an R-module (see Section 1).There has been a lot of effort already to find the graded Betti numbers of the Veronese ring S (d) . The problem can be really hard, namely, in [10], Ein and Lazarsfeld showed that for d ≫ 0 the graded Betti numbers β i,j (S (d) ) = 0 for many j if i is large: in particular, they proved that, for d ≫ 0 there exist l 1 , l 2 such that β p,p+q = 0 for all p in the range l 1 d q−1 ≤ p ≤ d+n n − l 2 d n−q . It is known (see references in [17]) that β i = β i,(i+1)d , for all i > 0, in the cases n = 2 or (d, n) = (2, 3). Instead, when d = 2 and n > 3, we have that the equality holds for i ≤ 5. In addition, for d = 2, all Betti numbers have been determined. In case n, d ≥ 3, Ottaviani and Paoletti (in [17])