The kinematics of mixing in two- and three-dimensional flows has been a much studied subject over the past two decades. Classical dynamical systems techniques have lead to a fairly detailed understanding of the structures that are responsible for chaotic advection in two-dimensional velocity fields with regular (periodic or quasi-periodic) time dependence. At the same time, turbulent flows with general time dependence and complex spatial structure remained inaccessible to these techniques. Further challenges have also remained unaddressed, including the finite-time nature of most mixing phenomena of interest, and the fact that in applications the velocity field is only available as a data set.

In this talk we survey recent results on mixing in finite-time velocity fields with general time dependence. We describe Lagrangian coherent structures that turn out to govern the stretching and folding observed in turbulent flows. We also discuss analytic and numerical results for the identification of such structures in numerical or experimental velocity fields. We illustrate the use of these methods on low-order models, simulations of geophysical turbulence, and laboratory experiments.