What do we know about self-similarity in fluid turbulence?

The evidence is reviewed on the statistical behavior of the small-scale fluctuations in high-Reynolds-number fluid turbulence. The qualitative phenomeno-logical information is summarized and the predictions of the 1941 Kolmogorov theory are reviewed. Then direct numerical simulation and its role in suggesting dynamical mechanisms are briefly discussed. Finally, the evidence on the multifractal structure of the dissipation field is reviewed. It is concluded that the experimental evidence for some kind of dynamical self-similarity is strong, but that there has been essentially no progress in fundamental theoretical understanding of the underlying mechanisms.