Richard Feynman's 1959 vision of controlling devices at small scales and swallowing the surgeon has inspired the science-fiction film and has played a crucial role in the rapid development of the microrobotics field. Sixty years later, we are currently witnessing a dramatic progress in this field, with artificial micro- and nanoscale robots moving within confined spaces, down to the cellular level, and performing a wide range of biomedical applications within the cellular interior while addressing the limitations of common passive nanosystems. In this review article, we discuss key recent advances in the field of micro/nanomotors toward important cellular applications. Specifically, we outline the distinct capabilities of nanoscale motors for such cellular applications and illustrate how the active movement of nanomotors leads to distinct advantages of rapid cell penetration, accelerated intracellular sensing, and effective intracellular delivery toward enhanced therapeutic efficiencies. We finalize by discussing the future prospects and key challenges that such micromotor technology face toward implementing practical intracellular applications. By increasing our knowledge of nanomotors' cell entry and of their behavior within the intracellular space, and by successfully addressing key challenges, we expect that next-generation nanomotors will lead to exciting advances toward cell-based diagnostics and therapy.