The evaluation of research to allocate government funding to universities is now common across the globe. "The Research Excellence Framework," introduced in the UK in 2014, marked a major change by extending assessment beyond the 'quality' of published research to include its real-world 'impact'. Impact submissions were a key determinant of the £4 billion allocated to universities following the exercise. The case studies supporting claims for impact are therefore a high stakes genre, with writers keen to make the most persuasive argument for their work. In this paper we examine 800 of these 'impact case studies' from disciplines across the academic spectrum to explore the rhetorical presentation of impact. We do this by analysing authors' use of hyperbolic and promotional language to embroider their presentations, discovering substantial hyping with a strong preference for boosting the novelty and certainty of the claims made. Chemistry and physics, the most abstract and theoretical disciplines of our selection, contained the most hyping items with fewer as we move along the hard/pure - soft/applied continuum as the real-world value of work becomes more apparent. We also show that hyping varies with the "type" of impact, with items targeting technological, economic and cultural areas the most prolific.