So I've just recently started learning path integral methods in QFT and string theory, and I've heard from numerous sources that the path integral (specifically fermionic path integrals, perhaps?) are objects which are not at all on solid mathematical ground. The feeling I get is that perhaps they're well enough understood for the kind of specific situations in physics, but lack generality. It seems Feynman discovered some phenomenal new mathematical landscape that no mathematicians had yet seen, much less understood. I'm wondering whether this is all accurate. And moreover, what about random surfaces in string theory? It seems there is very shaky mathematical rigour for integrating over topologies and embeddings.