Transport phenomena in fluid flows are observed ubiquitously in nature such as smoke rings in the air, pollutants in the aquifers, plankton blooms in the ocean, flames in combustion engines, and stirring a few drops of cream in a cup of coffee. We begin with examples of two dimensional Hamiltonian systems modeling incompressible planar flows, and illustrate the transition from ordered to chaotic flows as the Hamiltonian becomes more time dependent. We discuss diffusive, sub-diffusive, and residual diffusive behaviors, and their analysis via stochastic differential equation and a so called elephant random walk model. We then turn to level-set Hamilton-Jacobi models of the flames, and properties of the effective flame speeds in fluid flows under smoothing (such as regular diffusion and curvature) as well as stretching.