The surface accessible to the solvent and the volume contained within this surface are key notions both for the general description of proteins and other macromolecules, and for studying their interactions with the solvent and protein stability under a change of various parameters. There are many geometrical recipes for assigning a surface area and a volume to a protein. These recipes are not equivalent, and involve parameters whose definition is not always unique. We discuss the relative merits of the molecular and Voronoi definitions of protein volume, and their relation to experimentally measurable quantities like ''intrinsic compressibility.'' The molecular volume of a protein has the contradictory feature of not reflecting the packing of solvent molecules around it, being at the same time a function of the probe radius. The Voronoi volume does take into account modifications occurring in the solvent due to the presence of the protein and to possible external perturbations. Using the Voronoi volume results in a much better approximation of the experimental intrinsic compressibility. We suggest an explanation for the incorrect behavior of the molecular volume as a function of pressure, and propose using a pressure-dependent probe radius as a remedy in such cases when the molecular volume computation might be preferable.