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Measurements of the Kapitza conductance [sigma][sub [ital K]] made using a picosecond optical technique at temperatures between 50 and 300 K are presented for interfaces between metals and dielectrics. The Debye temperatures [ital FTHETA][sub [ital D]] of the metals (Pb, Au, Al, and Ti) were in the range from 102 to 426 K, while those of the dielectrics (BaF[sub 2], sapphire, and diamond) varied from 287 to 2200 K. Conductances measured between materials with [ital FTHETA][sub [ital D]] differing by less than about a factor of 5 were found to be in reasonable agreement with calculations based on a lattice dynamical theory. However, for more widely mismatched solids the measured conductances were found to be greatly in excess of the lattice dynamical calculations. In some of these cases the conductances exceeded even the phonon radiation limit, indicating that much of the heat flow between the solids was via an inelastic channel. It was demonstrated experimentally that the inelastic channel does not involve an interaction between the phonons in the dielectric and electrons in the metal. We show that the anomalously large conductance can be understood in terms of a model in which the anharmonicity of the metal plays an important role.