IAU Symposium 271 enhances our understanding of some of the important dynamical processes that are ubiquitous in many objects in the Universe, from stars like our Sun to galaxies. A wide range of temporal and spatial scales are generally present, so instabilities, highly nonlinear states and turbulence play a central role. Today, understanding the behaviour and evolution of such systems requires high-accuracy, multi-scale observations and thoughtful analysis of the data gathered, coupled with detailed theoretical study. High-performance numerical simulations have become an essential and revealing tool for assessing the often subtle highly nonlinear regime of such models. In IAU S271 world experts share their knowledge and perspectives on the latest advances in studying the common underlying processes from the field of nonlinear astrophysical dynamics. The symposium also celebrates the long and illustrious career of Professor Juri Toomre and his achievements in many realms of astrophysical dynamics.