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Recent high precision experimental data for heavy-ion fusion cross sectionsat energies in the vicinity of the Coulomb barrier systematically show that astrikingly large surface diffuseness parameter for a Woods-Saxon potential isrequired in order to fit the data. We discuss possible origins for thisanomaly, including the effects of dissipation and the sensitivity of fusioncross sections to the choice of inter-nuclear potential. Our study suggeststhat the frozen density approximation, which is often used in analyses ofheavy-ion reactions, may have to be re-examined for heavy-ion fusion.Comment: 12 pages, 6 eps figures, uses psfig.sty and sprocl.sty. A talk given  at the 4th Italy-Japan Symposium on Heavy-Ion Physics, Tokyo, Japan,  September 200

SURFACE DIFFUSENESS ANOMALY IN HEAVY-ION FUSION POTENTIALS