In recent years, there has been a steep rise in the quality of prostheses for patients with upper limb amputations. One common control method, using electromyographic (EMG) signals generated by muscle contractions, has allowed for an increase in the degrees of freedom (DOFs) of hand designs and a larger number of available grip patterns with little added complexity for the wearer. However, it provides little sensory feedback and requires non-natural control which must be learned by the user. Another recent improvement in prosthetic hand design instead employs electroneurographic (ENG) signals, requiring an interface directly with the peripheral nervous system (PNS) or the central nervous system (CNS) to control a prosthetic hand. While ENG methods are more invasive than using surface EMG for control, an interface with the PNS has the potential to provide more natural control and creates an avenue for both efferent and afferent sensory feedback. Despite the recent progress in design and control strategies, however, prosthetic hands are still far more limited than the actual human hand. This review outlines the recent progress in the development of EMG and ENG controlled prosthetic hands, discussing advancements in the areas of sensory feedback and control. The potential benefits and limitations of both control strategies, in terms of signal classification, invasiveness, and sensory feedback, are examined. A brief overview of interfaces with the CNS is provided, and potential future developments for these control methods are discussed.