Arguments are given which suggest that mass may be present in elliptical galaxies in the form of objects which lie within their Schwarxschild radii-black holes. The ohserved distribution of light and velocity dispersion of stars in the nuclear regions of bright ellipticals are then used to set limits to the amount of mass that can be present in this form. It is shown that in general the mass of a central hlack hole cannot exceed about 1010 M0, but that a much greater mass could be present in the form of many black holes with masses <102 1Io, in an extended distribution. To explain the high mass-to-light ratios and the observed run of surface brightness found in some ellipticals, the orbits of these subholes and the orbits of the visible stars must be mixed by encounterless relaxation. The rate at which a central black hole accelerates the evolution of a high-density stellar nucleus is calculated, and numerical results are given in a table. The form of the light distribution in the central few parsecs, if a massive black hole is present, is calculated. Only when a resolution <0'.' 1 is achieved for ellipticals in the Virgo cluster can direct tests of the presence of central black holes be made by ohserving the light distribution. Finally, a brief discussion is given of possible ways in which black holes may occur in ellipticals.