Let A {AN ELEMENT OF} gl(n,C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess (p, A) of the complete flag manifold consisting of those flags S_1, {SUBSET}... {SUBSET} S_{n-1}in C^n which satisfy the condition AS_i {SUBSET} S_{i+p} for all i. We show that if A has distinct eigenvalues, then Hess (p, A) is smooth and connected. The odd Betti numbers of Hess (p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers.