The idea of Responsible Research and Innovation (RRI) has become popular over the last decade to ensure that innovators take into account the social and ethical concerns that their innovations may raise, and organize their innovation processes so that they respond to the needs and values of society. In recent years, there has been increasing attention on assessing and measuring the RRI performance of innovators. A number of RRI assessment tools and measurement scales have been proposed. This contribution provides a conceptual framework to identify potential pitfalls of RRI measurement and assessment, and illustrates it on the basis of a number of RRI assessment approaches proposed in the literature. Two main categories of pitfalls are discussed, i.e. pitfalls in measuring RRI and pitfalls that arise from the strategic setting in which RRI assessment is applied. These two categories of pitfalls may reinforce each other. In order to navigate these pitfalls, a sharper distinction should be made between three underlying rationales for doing RRI assessment, i.e. learning, accountability and incentivizing. These underlying rationales correspond to different, and conflicting, requirements for RRI assessment methods.