The accurate timing and execution of organelle biogenesis is crucial for cell physiology. Centriole biogenesis is regulated by Polo-like kinase 4 (Plk4) and initiates in S-phase when a daughter centriole grows from the side of a preexisting mother. Here we show that Plk4 forms an adaptive oscillator at the base of the growing centriole to initiate and time centriole biogenesis, ensuring that centrioles grow at the right time and to the right size. The Plk4 oscillator is normally entrained to the cell-cycle oscillator, but can run autonomously of it – explaining how centrioles can duplicate independently of cell cycle progression under certain conditions. Mathematical modelling indicates that this autonomously oscillating system is generated...