Two tools from nonlinear dynamics recently advanced in oceanography, quasi-stationary Lagrangian coherent structures and time-asymptotic almost-invariant sets, are combined to diagnose the evolution of oil spills using climatological information of the state of ocean circulation in the Gulf of Mexico and of the oil degradation rate depending on its density. These represent the first steps toward developing a dynamical-systems-based oil spill prediction scheme. (Joint work with F. Beron-Vera and P. Miron; support provided by CONACyT–SENER.)