| Summary: | This empirical study explored how research can generate impacts by investigating different sorts of impacts from one academic field - mathematics - and the diverse mechanisms generating them. The multi-method study triangulated across: 1,2) content analysis of impact case studies and environment descriptions submitted to the UK Research Excellence Framework (REF) assessment; 3,4) a survey and focus group of heads of mathematics departments; 5) semi-structured interviews. Mathematics has had a full range of impact types, particularly conceptual impacts, although more tangible instrumental impacts were prioritised for REF. Multiple mechanisms were utilised, but seldom appeared in REF case studies. Long-term relationship-building and interdisciplinarity are particularly important. Departmental culture and certain knowledge intermediaries can play pro-active roles. <br/> In sharp contrast to simplistic linear narratives, we suggest that appreciation of diverse impact types, multiple, often informal, mechanisms and dynamic environments will enhance the likelihood of meaningful impacts being generated.
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