The human hand is a complex and versatile organ that enables humans to interact with the environment, communicate, create, and use tools. The control of the hand by the brain is a crucial aspect of human cognition and behaviour, but also a challenging problem for both neuroscience and engineering. The aim of this study is to review the current state of the art in hand and grasp control from a neuroscientific perspective, focusing on the brain mechanisms that underlie sensory integration for hand control and the engineering implications for developing artificial hands that can mimic and interface with the human brain. The brain controls the hand by processing and integrating sensory information from vision, proprioception, and touch, using different neural pathways. The user’s intention can be obtained to control the artificial hand by using different interfaces, such as electromyography, electroneurography, and electroencephalography. This and other sensory information can be exploited by different learning mechanisms that can help the user adapt to changes in sensory inputs or outputs, such as reinforcement learning, motor adaptation, and internal models. This work summarizes the main findings and challenges of each aspect of hand and grasp control research and highlights the gaps and limitations of the current approaches. In the last part, some open questions and future directions for hand and grasp control research are suggested by emphasizing the need for a neuroscientific approach that can bridge the gap between the brain and the hand.