The adiabatic single crystalline elastic constants of six Mg(Cu, Zn)/sub 2/ cubic Laves phases have been determined by the ultrasonic pulse-echo technique over the compositional range 10 to 50 mol % MgZn/sub 2/ and over the temperature range 4.2 to 300/sup 0/K. In addition the elastic constants of the hexagonal MgZn/sub 2/ Laves phase were measured as a function of temperature. A minimum exists in the compositional dependence of the cubic Mg(Cu, Zn)/sub 2/ phase at 20 mol % MgZn/sub 2/ for all three directly measured values and the anisotropy ratio shows a maximum at the same composition. These phenomena may be related to the fact that their Fermi surface touches the (311) faces of the Brillouin zones at this composition. The various energy contributions to the elastic shear constants were considered and it was concluded that the Fermi energy has to be included to achieve satisfactory agreement between the calculated and measured elastic constants. The Debye temperatures were calculated and were found to decrease at low zinc contents and then increase at compositions above 20 mol % MgZn/sub 2/ for the cubic Mg(Cu, Zn)/sub 2/ phase. The initial decrease is probably due to the substitution of the heavier zinc …