We study what makes a research grant application successful in terms of ability, type of research, experience, and demographics of the applicants. But our main objective is to investigate whether public funding organizations support the teams that are most likely to undertake transformative or "radical" research. Making use of the literature on recombinant innovation, we characterize such "radical teams" as those formed by eclectic and non-usual collaborators, and those that are heterogeneous and scientifically diverse. Our results, using data from the UK's Engineering and Physical Sciences Research Council (EPSRC), show that the more able, more basic, and more senior researchers, working in a top university, are more likely to be successful. But, radical teams are less likely to be funded by funding bodies. Our analysis of the research output of the awarded projects suggests that, voluntarily or involuntarily, the evaluation process in these organizations is biased against radical teams.