Models of randomly packed hard spheres exhibit some features of the properties of simple liquids, e.g. the packing density and the radial distribution. The value of the maximum packing density of spheres can be determined from models if care is taken to ensure random packing at the boundary surfaces and if correction is made for volume errors at the boundaries. Experiments for both the random 'loose' and the random close-packed densities are reported with + in. plexiglass, nylon and steel balls in air, and also with steel balls immersed in oil. A series of measurements for the random close-packed density has been made with up to 80 000 steel balls and with the aid of a mechanical vibrator. A computer analysis of the results permits a one-step, two-parameter extrapolation to infinite volume. The figure so obtained for the random close-packed density is 0.6366 k 0.0005, which represents an improvement in precision over previous results by an order of magnitude.