TITLE: Mathematics of path integral: state of the art
QUESTION [65 upvotes]: I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action) on some infinite-dimentional space of configurations (fields) of a system.
Unfortunately, it seems that, except for some few instances like Gaussian-type integrals, the quotation marks cannot be eliminated in the term "integration", cause a mathematically sound integration theory on infinite-dimensional spaces — I was told — has not been invented yet.
I would like to know the state of the art of the attempts to make this "path integral" into a well-defined mathematical entity.
Difficulties of analytical nature are certainly present, but I read somewhere that perhaps the true nature of path integral would be hidden in some combinatorial or higher-categorical structures which are not yet understood...
Edit: I should be more precise about the kind of answer that I expected to this question. I was not asking about reference for books/articles in which the path integral is treated at length and in detail. I'd have just liked to have some "fresh", (relatively) concise and not too-specialistic account of the situation; something like: "Essentially the problems are due to this and this, and there have been approaches X, Y, Z that focus on A, B, C; some progress have been made in ... but problems remain in ...".

REPLY [10 votes]: First, there are several rigorous definitions of integration in infinite dimensional spaces, like the Bochner integral in Banach spaces (see Wikipedia), or see the book by Parthasarathy: "Probability measures on metric spaces" (this includes the Gaussian probability measures used by constructive QFT already mentioned).
These cannot be used to make the Feynman path integral into a rigorous defined mathematical entity with finite values, i.e. the problem is to get an integral that spits out finite numbers in physically interesting models.
For starters, there cannot be a translationally invariant measure (on the Borel sigma algebra) other than the one that assigns infinite volume to every open set in an infinite dimensional metric space (hint: a ball of radius r contains infinitly many pairwise disjunct copies of the ball of radius r/2). So the path integral, as it is written by physicists, certainly has no interpretation via a translationally invariant measure, contrary to what the notation usually employed may suggest.
While there currently is no mathematically rigorous definition of a Feynman path integral applicable to an interesting subset of physical models, here are some books that give some hints at the current state of the affair:
Huang and Yan: "Introduction to Infinite Dimensional Stochastic Analysis" (this contains a description of the Feynman path integral from the viewpoint of "white noise analysis"),
Sergio Albeverio, Raphael Hoegh-Krohn; Sonia Mazzucchi:"Mathematical theory of Feynman path integrals. An introduction",
Pierre Cartier, Cecile DeWitt-Morette: "Functional integration: action and symmetries".
BTW: This is in a certain sense a "one million dollar" question because one of the millenium problems of the Clay Mathematics Institute is a rigorous construction of Yang-Mills theories.