The advection of tracer particles in hydrodynamical flows represents one of the successful applications of chaos theory. The basic observation is that molecular diffusion is negligible on the typical time scale of the flow. As a result, in the absence of any diffusion-enhancing mechanism such as hydrodynamical turbulence, advection dominates. Indeed, the main physical mechanism for fluid stirring is advection, whose efficiency can be enhanced greatly by chaotic dynamics. The spreading of pollutants on large scales is also dominated by advection. Potential applications of chaotic advection range from laboratory investigations of fluid dynamics to the study of large-scale environmental flows. From the point of view of dynamical systems, an appealing feature of the passive advection problem is that its phase space coincides with the physical space of the fluid, rendering possible direct experimental observation and characterization of fractal structures associated with chaotic dynamics.