Neuromorphic hardware enables fast and power-efficient neural network-based artificial intelligence that is well suited to solving robotic tasks. Neuromorphic algorithms can be further developed following neural computing principles and neural network architectures inspired by biological neural systems. In this Viewpoint, we provide an overview of recent insights from neuroscience that could enhance signal processing in artificial neural networks on chip and unlock innovative applications in robotics and autonomous intelligent systems. These insights uncover computing principles, primitives, and algorithms on different levels of abstraction and call for more research into the basis of neural computation and neuronally inspired computing hardware.