Computational modelling of biological systems is becoming increasingly important in efforts to better understand complex biological behaviours. In this review, we distinguish between two types of biological models—mathematical and computational— which differ in their representations of biological phenomena. We call the approach of constructing computational models of biological systems ‘executable biology’, as it focuses on the design of executable computer algorithms that mimic biological phenomena. We survey the main modelling efforts in this direction, emphasize the applicability and benefits of executable models in biological research and highlight some of the challenges that executable biology poses for biology and computer science. We claim that for executable biology to reach its full potential as a mainstream biological technique, formal and algorithmic approaches must be integrated into biological research. This will drive biology toward a more precise engineering discipline.