We study cubature formulas for d-dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree the number of knots depends on the dimension in an order-optimal way; for fixed dimension the dependence on the degree is almost optimal. In addition, the cubature formulas are universal: their error is almost optimal in two different scales of function spaces. The construction is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas.
Constructive Approximation, to appear.