The de Haas-van Alphen effect in aluminium has been studied by measuring the oscillatory variation with magnetic field of the torque on a single crystal at liquid-helium temperatures in fields up to 15$\cdot $4 kG. The torques were measured by an electronic feed-back device designed to reduce the twisting motion of the crystal during torque measurement. Particular attention was paid to the variation of the periods of the relatively high-frequency components of the oscillations with the orientation of the field relative to the crystal axes. A consistent interpretation was obtained by supposing that there are three such periodic components for each field direction, though often the relative amplitude of one or two of these is negligibly small. In terms of Onsager's theory, the period is inversely proportional to the maximum area of cross-section of the Fermi surface by planes normal to the field, and the three periodic components have been shown to correspond to three identical cushion-shaped pieces of the Fermi surface with their principal axes mutually perpendicular. The location of these pieces in relation to the Brillouin zone is discussed and the characteristic dimensions of each piece have been calculated. Some results on the variation of the period of the relatively low-frequency component with field orientation in a (100) plane are described, but no detailed interpretation in terms of the shape of appropriate parts of the Fermi surface has been obtained. A few results on the variation of the oscillation amplitude with field orientation and on the temperature-dependence of amplitude are also presented. Alloying the aluminium with up to 0$\cdot $26% magnesium increases the period of the low-frequency component by about 2$\cdot $3%; this suggests that this component arises from electrons rather than holes. The period of the high-frequency component is not significantly changed by alloying.