We use a self-consistent model of galaxy formation and the evolution of the intergalactic medium to study the effects of the reionization of the Universe at high redshift on the properties of satellite galaxies like those seen around the Milky Way. Photoionization suppresses the formation of small galaxies, so that surviving satellites are preferentially those that formed before the Universe reionized. As a result, the number of satellites expected today is about an order of magnitude smaller than the number inferred by identifying satellites with subhaloes of the same circular velocity in high-resolution simulations of the dark matter. The resulting satellite population has an abundance similar to that observed in the Local Group, although the distribution of circular velocities differs somewhat from the available data. We explore many other properties of satellite galaxies, including their gas content, metallicity and star formation rate, and find generally good agreement with available data. Our model predicts the existence of many as yet undetected satellites in the Local Group. We quantify their observability in terms of their apparent magnitude and surface brightness, and also in terms of their constituent stars. A near-complete census of the Milky Way's satellites would require imaging to V≈20 and to a surface brightness fainter than 26 V-band magnitudes per square arcsecond. Satellites with integrated luminosity V=15 should contain of order 100 stars brighter than B=26, with central stellar densities of a few tens per square arcminute. Discovery of a large population of faint satellites would provide a strong test of current models of galaxy formation.