All dynamical motion of matter is accompanied by fluctuations. While these fluctuations are often suppressed, e.g. by cooling, for both weakly and strongly damped systems they take a fundamental role in the presence of external driving and intrinsic nonlinearities. By controlling effects like nonlinear damping, anharmonic potentials or tailored optimal driving, the systems enter new dynamical regimes which will be the central subjects of our investigations. Synergistically combining underdamped mechanics, overdamped colloidal suspensions, and mathematical modelling, we will explore this uncharted territory in classical and quantum matter.