Enormous capture cross sections in the range 10^-15 cm2 to 10^-12 cm2 have been observed for a wide variety of Coulomb attractive centers in Si and Ge, some involving binding energies many times the Debye energy. Whereas multiphonon transitions to the ground state yield cross sections five to ten orders of magnitude too small, capture into excited states of large radius followed by a cascade of one-phonon transitions leads to cross sections of the right order of magnitude. The initial capturing event is likely to involve an optical phonon or an intervalley collision in the room temperature range, but the acoustic phonon contribution will predominate at low temperatures.