High-pressure experimental data are used to evaluate the validity of the Lindemann melting criterion for minerals. Because anharmonicity can be significant at the high temperatures associated with melting, leading order anharmonic contributions to the mean-square atomic vibrations have been included in the present lattice dynamical formulation of the Lindemann criterion. The parameters required in this Lindemann expression can be determined from an inversion of thermodynamic data so that a direct comparison with experimental melting data can be made without resort to model potentials. In general, we find that although atomic vibrations can be signifiantly anharmonic near the melting point, the shape of the melting curve predicted from the Lindemann criterion is not much influenced by anharmonicity. We demonstrate that the Lindemann expression is inconsistent with experimental melting data for a variety of substances, particularly at low pressures. In some cases (argon, sodium chloride and potassium chloride), the melting slopes observed at high pressures are consistent with a Lindemann expression. In general, we find that a Lindemann criterion is not an adequate description of the melting behavior of minerals. Therefore, an application of the Lindemann law to obtain the melting temperatures of minerals at high pressurs, and to determine the fusion curve of the earth's deep interior is, at present, speculative.