From rain drops landing on the ocean to inkjet printing, the impact of droplets onto moving liquid surfaces is a ubiquitous process in nature and in industry. A rich range of phenomena can arise. The behaviour depends on the inertia, the properties of the drops and the relative speeds in the impact zone. While the result ranges from tranquil coalescence to violent splashing, intermediate regimes also occur, including partial and complete bouncing and even 'surfing' of the droplet. These regimes are determined by the ratio of the drop and surface velocities and the liquid properties. A regime diagram can be constructed in which distinct dynamical regimes are clearly identified.