The multifractal formalism for the eulerian statistics of small-scale dynamics in turbulent flows is reviewed. Theoretical extensions of these results (the statistics of small volume averages of the energy dissipation rate) are used to predict properties of the probability distribution of the local energy dissipation rate at a fixed point. The improved parametrization of the eulerian statistics allows the lagrangian statistics (those for a fixed fluid particle in contrast to the eulerian statistics at a fixed point) to be determined exactly by using results derived as a consequence of incompressibility. Several properties of particle trajectories in a turbulent flow can be predicted with these new lagrangian statistics. In particular, a trajectory is typically smooth and generally unremarkable in its features. This contrasts the often suggested description: that of a highly convoluted and intricately structured `fractal' curve. Some of the traditional dispersion results, which depend on the lagrangian statistics, are shown to be only weakly influenced by the intermittency inherent in the multifractal character of turbulence.