The rapid development of AI-based technologies significantly impacts almost all human activities as they are tied to already existing underlying systems and services. In order to make sure that these technologies are at least transparent if not provably beneficial for human beings and society and represent a true progress, AI governance will play a key role. In this paper, we propose to reflect on the notion of ‘digital responsibility’ to account for the responsibility of economic actors. Our objective is to provide an outline of what digital responsibility is and to propose a Digital Responsibility Index to assess corporate behavior. We argue that a Digital Responsibility Index can play a central role in restoring trust in a datadriven economy and create a virtuous circle, contributing to a sustainable growth. This perspective is part of AI governance because it provides a concrete way of quantifying the implementation of AI principles in corporate practice.