In physics I have come across contexts where apparently path integrals are well-defined, and others where they are not. However I have no clear understanding of when and why they succeed or fail to be well-defined.
Question 1. What are the obstructions to path integrals being well-defined? When path integrals are well-defined, what goes ‘right’ which otherwise typically goes wrong?
I have seen other questions on here that are similar, but I have never really understood the answers. For instance often approaches involving the limit of a discretization lead to divergences. Can a regularization scheme not be included in the path integral definition, to avoid this?
Even if this does not work, are there no other integration measures over path or configuartion spaces which give sensible answers? Or do appropriate measures only exist in special cases?
Question 2. Is there a simple summary of situations where we currently know path integrals are well-defined?
For instance, this question appears to indicate that path integrals with Euclidean action over paths in $\mathbb{R}^n$ are always well defined.
Question 3. I have heard path integrals are always well-defined for TQFTs. If this is true, then why?