We show that an arbitrary body or aggregate can be made perfectly absorbing at discrete frequencies if a precise amount of dissipation is added under specific conditions of coherent monochromatic illumination. This effect arises from the interaction of optical absorption and wave interference and corresponds to moving a zero of the elastic S matrix onto the real wave vector axis. It is thus the time-reversed process of lasing at threshold. The effect is demonstrated in a simple Si slab geometry illuminated in the 500-900 nm range. Coherent perfect absorbers act as linear, absorptive interferometers, which may be useful as detectors, transducers, and switches.