Artificial, self-propelled micro- and nanomotors are small devices capable of autonomous movement, which are a powerful scientific innovation for solving various medical and environmental issues. Their design is frequently inspired by complex biological structures which are composed of biopolymers and their composites. The choice of materials for nano- and micromachines is crucial for their shape, mechanism and efficiency of propulsion. In this review, we discuss the utilization and fabrication of polymers as soft components of micro- and nanomotors.