The behavior of diffusion coefficients in the critical region is important to understanding transport phenomena in supercritical fluids. For example, it is often claimed that mass transfer rates are high in the critical region. However, it is also generally accepted that diffusion processes slow down dramatically as the critical point is approached, although there appear to be conflicting results on this point in the literature. From published experimental data for the self-diffusion constant of pure fluids in the critical region one could conclude that this property: (1) approaches zero at the critical point (Cini-Castagnoli et al. Physica 48 (1970) 153), (2) remains finite there, showing no anomalous behavior (Etesse et al. Physica B 183 (1993) 45) or (3) displays singular behavior, approaching infinity, at the critical point itself (Duffield and Harris, Ber.Bunsenges-Gesellschaft 80 (1976) 157). We discuss these and related issues from the vantage point of dynamic scaling theory and the Onsager regression hypothesis. We conclude that while self-diffusion coefficients are not predicted to show anomalous behavior at the critical point. consistent with much of the available data, Fickian (transport) diffusion coefficients are predicted to approach zero there. We are unaware of any measurements of this latter property in a pure fluid consistent with scaling predictions.