This paper presents a system for the execution of autonomous cinematography missions with a team of drones. The system allows media directors to design missions involving different types of shots with one or multiple cameras, running sequentially or concurrently. We introduce the complete architecture, which includes components for mission design, planning and execution. Then, we focus on the components related to autonomous mission execution. First, we propose a novel parametric description for shots, considering different types of camera motion and tracked targets; and we use it to implement a set of canonical shots. Second, for multi-drone shot execution, we propose distributed schedulers that activate different shot controllers on board the drones. Moreover, an event-based mechanism is used to synchronize shot execution among the drones and to account for inaccuracies during shot planning. Finally, we showcase the system with field experiments filming sport activities, including a real regatta event. We report on system integration and lessons learnt during our experimental campaigns.