Funding organisations are increasingly asking academics to show evidence of the economic and social value generated by their research. These requests have often been associated with the emergence of a so-called "new social contract for research" and are related to the implementation of new research evaluation systems. Although the research evaluation rhetoric is similar across countries and organisations, in practice evaluation can fulfil very different purposes. Additionally, the assessment of the public value of research poses different challenges depending on the academic field under analysis. This paper distinguishes three main research evaluation goals: to inform the distribution of public resources among competing objectives or performers, to help improve the implementation of policies and programmes, and to control the use of public funds. It then argues that assessing the value of research in the arts and humanities calls for a research methodology capable of providing a fine-grained understanding of the variety of, often diffuse, ways in which arts and humanities research can generate value. The methods that we need to do this are better suited to fulfil the improvement goal of evaluation, and require a "formative" approach to evaluation supporting the social engagement of academic researchers.