Basins of attraction are complex, beautiful, and deeply intriguing objects in nonlinear dynamics. My earlier work explored how basins of attraction, together with their boundaries, inherit the symmetries of the governing equations. One interesting observation is that basin boundaries may possess higher symmetry than the basins themselves. At present, I am studying locally coupled Kuramoto models, where we have found that the proportions of different basins of attraction appear to be Gaussian distributed. We are now trying to develop a theoretical proof of this observation. All of these projects are ongoing.