Yesterday our institute hosted Helen Fay Dowker. She gave a talk at our traditional thursday lunch meetings and the title was something like the path integral interpretation of quantum mechanics.
Fay gave an introduction about Dirac's 1932 paper that the lagrangian approach to classical mechanics was probably more fundamental than the hamiltonian approach because the former is relativistically invariant whereas the latter is "essentially nonrelativistic." This is because in the definition of hamiltonian time is an exceptional dimension and the evolution occurs in that dimension. This breaks the equality between time and space that general relativity is based on.
In a path integral we sum over histories from which the physical world is described directly in terms of events in spacetime. The path integral approach, however,has been championed in more recent times by Jim Hartle in a probabilistic nature and by Raphael Sorkin in non-probabilistic nature.
This path integral talk would have been tasted better with noodles, of course!
And that is my memo.