The invention concerns alumina-based spherically shaped grains produced from pastes of moistened particles. These grains are produced by subjecting the pastes, in the form of grains of any shape whatever, to the action of mechanical vibrations of frequencies within the range of from a few herz, in liquids which are slightly or not at all water-miscible. To the alumina, which is the main component, particles of various natures and acids, bases and various compounds which are dissolved in the water used for preparing the pastes, can be added. Once matured, dried and calcined, the grains have a very high level of resistance to attrition, a porosity comprising pores of large dimensions, and can be used in particular for adsorption, chromatography and catalysis.