The author discusses the problem of relating the fundamental elastic properties of anisotropic single crystals to those of polycrystalline bulk materials. It is shown that the orientation-independent relationships are not in fact independent of each other and are insufficient to enable the constants to be found. Some useful results do, however, follow. Thus, the methods of Reuss (1929), Voigt (1928), and Hill (1952), lead to the same calculation of bulk modulus for polycrystalline materials and show that it should be identical with the single-crystal value. It is also shown why polycrystalline cubic metals behave in an isotropic manner when the grains are completely randomly oriented. Departures from randomness (development of an orientation texture) render increasingly invalid the widely accepted relationship between the moduli, E and G, and v (Poisson's ratio), viz. E=2G(1+v).