Biological molecular motors (or biomolecular motors for short) are nature's solution to the efficient conversion of chemical energy to mechanical movement. In biological systems, these fascinating molecules are responsible for movement of molecules, organelles, cells, and whole animals. In engineered systems, these motors can potentially be used to power actuators and engines, shuttle cargo to sensors, and enable new computing paradigms. Here, we review the progress in the past decade in the integration of biomolecular motors into hybrid nanosystems. After briefly introducing the motor proteins kinesin and myosin and their associated cytoskeletal filaments, we review recent work aiming for the integration of these biomolecular motors into actuators, sensors, and computing devices. In some systems, the creation of mechanical work and the processing of information become intertwined at the molecular scale, creating a fascinating type of "active matter". We discuss efforts to optimize biomolecular motor performance, construct new motors combining artificial and biological components, and contrast biomolecular motors with current artificial molecular motors. A recurrent theme in the work of the past decade was the induction and utilization of collective behavior between motile systems powered by biomolecular motors, and we discuss these advances. The exertion of external control over the motile structures powered by biomolecular motors has remained a topic of many studies describing exciting progress. Finally, we review the current limitations and challenges for the construction of hybrid systems powered by biomolecular motors and try to ascertain if there are theoretical performance limits. Engineering with biomolecular motors has the potential to yield commercially viable devices, but it also sharpens our understanding of the design problems solved by evolution in nature. This increased understanding is valuable for synthetic biology and potentially also for medicine.