Keywords: passive scalar, enhanced diffusion, mixing properties Abstract: We consider passive scalar equation on a compact domain or manifold. The fluid flow can aid diffusion and increase the speed of convergence of the initial distribution to its average. We consider either stationary or time periodic flows, and derive a sharp characterization of flows that are particularly effective in enhancing the relaxation speed to mean value. The characterization links enhanced relaxation with spectral properties of the dynamical system generated by flow. The results also provide an indication that time dependence of the flow may improve relaxation enhancing properties. Methods used involve a mix of PDE techniques and functional analysis. A key role is played by estimates similar to ones used in quantum dynamics to measure the rate of wavepacket propagation. The talk is based on works joint with P. Constantin, L. Ryzhik, R. Shterenberg and A. Zlatos.