Long-term changes in knowledge production can produce mismatches between the research that society requires and the research that society produces — what we term ‘relevance gaps’. This paper explores what can be done to close them. The paper argues that current structures for governing research are often inappropriate, damage the reputation and value system of the academy, and produce a widespread perception that much research is irrelevant. New ways are needed to address how disciplinary value judgements and the structure of peer review influence the direction of academic research. Alternatives to current peer-review practices and guidelines for funding agencies are proposed.