Bioinspired out-of-equilibrium systems will set the scene for the next generation of molecular materials with active, adaptive, autonomous, emergent and intelligent behavior. Indeed life provides the best demonstrations of complex and functional out-of-equilibrium systems: cells keep track of time, communicate, move, adapt, evolve and replicate continuously. Stirred by the understanding of biological principles, artificial out-of-equilibrium systems are emerging in many fields of soft matter science. Here we put in perspective the molecular mechanisms driving biological functions with the ones driving synthetic molecular systems. Focusing on principles that enable new levels of functionalities (temporal control, autonomous structures, motion and work generation, information processing) rather than on specific material classes, we outline key cross-disciplinary concepts that emerge in this challenging field. Ultimately, the goal is to inspire and support new generations of autonomous and adaptive molecular devices fueled by self-regulating chemistry.