CONTEXT: Cosmic voids are observed in the distribution of galaxies and, to some extent, in the dark matter distribution. If these distributions have fractal geometry, it must be reflected in the geometry of voids; in particular, we expect scaling sizes of voids. However, this scaling is not well demonstrated in galaxy surveys yet. AIMS: Our objective is to understand the geometry of cosmic voids in relation to a fractal structure of matter. We intend to distinguish monofractal voids from multifractal voids, regarding their scaling properties. We plan to analyse voids in the distributions of mass concentrations (halos) in a multifractal and their relation to galaxy voids. METHODS: We make a statistical analysis of point distributions based on the void probability function and correlation functions. We assume that voids are spherical and devise a simple spherical void finder. For continuous mass distributions, we employ the methods of fractal geometry. We confirm the analytical predictions with numerical simulations. Smoothed mass distributions are suitable for the method of excursion sets. RESULTS: Voids are very nonlinear and non-perturbative structures. Voids reflect the fractal geometry of the matter distribution but not always directly: scaling sizes of voids imply fractal geometry, but fractal voids may have a complicated geometry and may not have scaling sizes. Proper multifractal voids are of this type. A natural multifractal biasing model implies that the voids in the galaxy distribution inherit the same complicated geometry. CONCLUSIONS: Current galaxy surveys as well as cosmological N-body simulations indicate that cosmic voids are proper multifractal voids. This implies the presence in the voids of galaxies or, at least, small dark matter halos.