TITLE: Kuranishi structures vs polyfolds
QUESTION [10 upvotes]: Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like to at least associate a fundamental class to the moduli space in question.
It looks like two approaches dominate: Kuranishi structures and polyfolds.
Both seem to be mammoth projects. Before diving seriously into one of them I may ask:
What are the advantages/drawbacks of these approaches? What is their motivation? Are they rigourosly proved? Are there reasonable alternatives?

REPLY [4 votes]: Perhaps you would like to take a look at a recent preprint by McDuff and Wehrheim 
http://arxiv.org/abs/1208.1340
This rather long paper seeks to explain the topological and analytic issues involved and provide the beginnings of a framework for resolving them.
These authors are well known for their attention to detail and expository ability, so it might be worthwhile to take a look.
I've heard rumors that an eventual Polyfold approach and the one taken by McDuff and Wehrheim share quite a bit of overlap in the end, but am not enlightened enough to say more on this matter.