Neuromorphic computing takes inspiration from the brain to create energy efficient hardware for information processing, capable of highly sophisticated tasks. In this article, we make the case that building this new hardware necessitates reinventing electronics. We show that research in physics and material science will be key to create artificial nano-neurons and synapses, to connect them together in huge numbers, to organize them in complex systems, and to compute with them efficiently. We describe how some researchers choose to take inspiration from artificial intelligence to move forward in this direction, whereas others prefer taking inspiration from neuroscience, and we highlight recent striking results obtained with these two approaches. Finally, we discuss the challenges and perspectives in neuromorphic physics, which include developing the algorithms and the hardware hand in hand, making significant advances with small toy systems, as well as building large scale networks.