The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes. With e 1 , . . . , e n denoting the standard basis vectors of R n and letting x k = e 1 + · · · + e k (k = 1, 2, . . . , n), the setis the vertex set of a generalized regular hyperoctahedron in R n . A finite set X ⊂ R n with a weight function w :holds for every polynomial f of total degree at most t; here R is the set of norms of the points in X , W r is the total weight of all elements of X with norm r , S r is the n-dimensional sphere of radius r centered at the origin, andf S r is the average of f over S r .Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum t for which it is a Euclidean design) equal to 3, 5, or 7. We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7. In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for t = 5, a set of three equations for t = 7, and a set of seven equations for t = 9.Springer 376 J Algebr Comb (2007) 25: Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fishertype inequality |X | ≥ N (n, p, t) for the minimum size of a Euclidean t-design in R n on p = |R| concentric spheres (assuming that the design is antipodal if t is odd). A Euclidean design with exactly N (n, p, t) points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for N (n, p, t) = (3, 2, 5), (3, 3, 7), and (4, 2, 7).