We discuss the results of extensive numerical simulations in order to estimate the scaling exponents associated with kinetic roughening in higher dimensions, up to d=7+1. To this end, we study the restricted solid - on - solid growth model, for which we employ a novel fitting {\it ansatz} for the spatially averaged height correlation function $\bar G(t) \sim t^{2β}$ to estimate the scaling exponent $β$. Using this method, we present a quantitative determination of $β$ in d=3+1 and 4+1 dimensions. To check the consistency of these results, we also compute the interface width and determine $β$ and $χ$ from it independently. Our results are in disagreement with all existing theories and conjectures, but in four dimensions they are in good agreement with recent simulations of Forrest and Tang [{\it Phys. Rev. Lett.} {\bf 64}:1405 (1990)] for a different growth model. Above five dimensions, we use the time dependence of the width to obtain lower bound estimates for $β$. Within the accuracy of our data, we find no indication of an upper critical dimension up to d=7+1.

17 pages, TEX, figures upon request from ala@phcu.helsinki.fi