This Letter summarizes numerical results from [J. Comp. Phys. (to appear)] which show that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like h"k, ' ' " 2, Here, K-epsilon denotes the subset of the classical energy surface {H = E} which stays bounded for all time under the flow of H and D(K-E) denotes its fractal dimension, Since the number of bound states in a quanturn systern with n degrees of freedom scales like h (n). this suggests that the quantity (D(K-epsilon) + 1)/2 represents the effective number of degrees of freedom in chaotic scattering problerns. (C) 2002 Published by Elsevier Science B.V.