The term 'synergy' - from the Greek synergia - means 'working together'. The concept of multiple elements working together towards a common goal has been extensively used in neuroscience to develop theoretical frameworks, experimental approaches, and analytical techniques to understand neural control of movement, and for applications for neuro-rehabilitation. In the past decade, roboticists have successfully applied the framework of synergies to create novel design and control concepts for artificial hands, i.e., robotic hands and prostheses. At the same time, robotic research on the sensorimotor integration underlying the control and sensing of artificial hands has inspired new research approaches in neuroscience, and has provided useful instruments for novel experiments. The ambitious goal of integrating expertise and research approaches in robotics and neuroscience to study the properties and applications of the concept of synergies is generating a number of multidisciplinary cooperative projects, among which the recently finished 4-year European project "The Hand Embodied" (THE). This paper reviews the main insights provided by this framework. Specifically, we provide an overview of neuroscientific bases of hand synergies and introduce how robotics has leveraged the insights from neuroscience for innovative design in hardware and controllers for biomedical engineering applications, including myoelectric hand prostheses, devices for haptics research, and wearable sensing of human hand kinematics. The review also emphasizes how this multidisciplinary collaboration has generated new ways to conceptualize a synergy-based approach for robotics, and provides guidelines and principles for analyzing human behavior and synthesizing artificial robotic systems based on a theory of synergies.