  Feynman path integrals are now a standard tool in quantum physics and their
use in differential geometry leads to new mathematical insights. A logical
treatment of quantum phenomena seems to require a sustained mathematical
analysis of path integrals. The subject is complicated by the fact that their
application in flat space-time is quite different from how path integrals are
used in, say, topological quantum field theory, where there is no natural
notion of time translation. An historical background is given in this paper and
a few mathematical approaches to Feynman path integrals in the context of
nonrelativistic quantum mechanics and scalar quantum fields with polynomial
self-interactions are outlined.
