It's probably needed to mention that I'm still a graduate student, so I studied both General Relativity and QFT but gravity was never mentioned in my QFT courses, nor the other way around. Nevertheless, from some independent study, it seems to me that the next step towards a TOE is always said to be a quantized theory of gravity, the search for a graviton, etc.
My question is, couldn't the other way be useful to analyze as well? Has anyone tried to give a curved manifold representation of QFT? If so, has it failed? If not, how come we don't hear about it more often?