The determination of the low-frequency (typically 0-1 MHz) dielectric dispersion of colloidal suspensions may become an electrokinetic tool of wider use if the accuracy of experimental data can be improved and if trustable theories, available for a wide range of situations, are made available. In the present work, we focus on the latter aspect: Since the dielectric constant of the suspensions is in fact a collective property, its determination could be most useful in concentrated suspensions. This is our aim in this paper. Using the classical electrokinetic equations and a cell model accounting for particle-particle interactions, we present calculations of the dielectric spectra of concentrated (volume fractions up to 50%) suspensions of spheres. Most of our results cannot be thought of as any sort of extrapolation of those corresponding to dilute suspensions (the reverse is true), and in fact the notion of a dilute colloidal system is itself not free of uncertainties, as no "critical volume fraction" can be identified separating the dilute and concentrated ranges. According to the calculations described, increasing the particle concentration by a sufficient amount can lead to a decrease of the dielectric constant of the whole system that can be well below that of the dispersion medium, even for high zeta potentials, zeta. The latter quantity affects (and this is also true if phi --> 0) considerably both the dielectric constant epsilon(r)' and the relaxation frequency, f(rel): When zeta is increased, both the low-frequency value, epsilon(r)'(0), of epsilon(r)', and f(rel) increase at all particle concentrations. We also analyze the effect of the product kappaa, where a is the particle radius and kappa is the reciprocal Debye length: higher kappaa values correspond to larger epsilon(r)'(0) and lower f(rel). Finally, the model is compared to previously reported experimental data: it is found that the qualitative agreement is excellent both concerning epsilonr'(0) and f(rel). Possible improvements of the theory, particularly the inclusion of a dynamic Stern layer, are suggested. (C) 2003 American Institute of Physics.