Fluid turbulence has attracted the attention of physicists, mathematicians, and engineers for over 100 years, yet it remains an unsolved problem. The reasons for the difficulties are outlined and recent advances in describing its small-scale statistical structure are described. Contrary to traditional notions, new experimental evidence indicates that the small scales are anisotropic, reflecting the overall character of the flow. The consequences of this finding with regard to the long-held postulate of the universality of the small-scale turbulence structure are discussed.