In the paper, the author takes stock of the conceptual reflection and empirical studies described in the current scientific literature on responsible innovation in the context of the emergence of the Responsible Research and Innovation (RRI) concept. RRI has been promoted in the European Union as a part of the Europe 2020 strategy with the objective of making research and innovation more sustainable and inclusive. As more than half of the EU’s firms declare they are conducting innovation activities, RRI problems are more relevant than ever. There remain many open questions, unresolved dilemmas and empirical white spots that call for more research in this field. This paper’s focus is the problem of RRI acceptance as a global framework for responsible innovation and the scarcity of suitable instruments that may help industries understand and adopt this concept. The main contributions of this paper include critical analysis of the RRI concept and its implications for industry, proposing a concept of an RRI index for innovating enterprises.