Innovations in biological evolution and in technology have many common features. Some of them involve similar processes, such as trial and error and horizontal information transfer. Others describe analogous outcomes such as multiple independent origins of similar innovations. Yet others display similar temporal patterns such as episodic bursts of change separated by periods of stasis. We review nine such commonalities, and propose that the mathematical concept of a space of innovations, discoveries or designs can help explain them. This concept can also help demolish a persistent conceptual wall between technological and biological innovation.