The astrophysical rapid neutron capture process or `$r$ process' of nucleosynthesis is believed to be responsible for the production of approximately half the heavy element abundances found in nature. This multifaceted problem remains one of the greatest open challenges in all of physics. Knowledge of nuclear physics properties such as masses, $\beta$-decay and neutron capture rates, as well as $\beta$-delayed neutron emission probabilities are critical inputs that go into calculations of $r$-process nucleosynthesis. While properties of nuclei near stability have been established, much still remains unknown regarding neutron-rich nuclei far from stability that may participate in the $r$ process. Sensitivity studies gauge the astrophysical response of a change in nuclear physics input(s) which allows for the isolation of the most important nuclear properties that shape the final abundances observed in nature. This review summarizes the extent of recent sensitivity studies and highlights how these studies play a key role in facilitating new insight into the $r$ process. The development of these tools promotes a focused effort for state-of-the-art measurements, motivates construction of new facilities and will ultimately move the community towards addressing the grand challenge of `How were the elements from iron to uranium made?'.