The two- and three-dimensional random packing problems in Palásti's (1960) sense are studied by means of Monte Carlo simulations and extrapolation techniques. In the two-dimensional case, a 95% confidence interval of 0·5629 ± 0·0006 is obtained for the limit of mean random packing density. This estimate is slightly but significantly higher than the value conjectured by Palásti, c2 ≏0·5589. For the three-dimensional case, we calculate only a lower bound of the limiting density and give a 95% confidence interval of 0·4148 ± 0·0025, which is also fairly near to the conjectured value ( c3 ≏0·4178).