How are infinite-dimensional manifolds most commonly treated? I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for an answer to (mainly the first part of) my question.
At a summer school I recently attended, infinite-dimensional manifolds popped up. I have never worked with them before (although I'm very familiar with finite-dimensional manifolds). The lecturer at the school did not give any details about the technical realization of infinite-dimensional manifolds, mentioning that there were issues (such as picking a topology) that he would leave out for the sake of clarity, since the relevant results were true independent of the exact technical details. An internet search reveals that Banach manifolds are one way of treating infinite-dimensional manifolds, but there are others.

Are Banach manifolds the most common way of defining infinite-dimensional manifolds, or are there other notions commonly used? Is there a more or less universal consensus about when to use which treatment? What are the most important (dis)advantages of each?
Supposing I want to learn the basics of infinite-dimensional manifolds, are there any well-written introductory texts you would recommend? (on StackExchange, The Convenient Setting of Global Analysis by A. Kriegl and P. Michor was recommended)

 A: Banach manifolds  have found many uses such as,  gauge theory (Donaldson theory, Seiberg-Witten theory, Floer theory), symplectic topology (Gromov-Witten theory), to name few I am more familiar with.
One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach context  takes a simpler form, and thus Banach manifolds are easier to recognize.
One disadvantage of Banach manifolds over  Frechet manifolds is the fact that natural  notions of  real analyticity   are harder to implement on Banach spaces.
One place to learn about Banach manifolds is Lang's Differential and Riemannian Manifolds
A: One emerging trend seems to be that the category
of sheaves of sets on the site of smooth manifolds
(also known as the category of generalized manifolds)
is the right category of what one might call smooth sets.
(Here we no longer restrict our attention to spaces that look
the same at every point, and in fact we have spaces
that have no points at all.)
In particular, it includes all sorts of infinite-dimensional manifolds, such as Banach and Fréchet manifolds
as full subcategories.
It also contains many other categories of smooth
objects, such as diffeological spaces, as full subcategories.
Also, this category can be generalized nicely to higher smooth homotopy
types, e.g., smooth stacks = smooth homotopy 1-types,
which constantly pop up even if you're studying ordinary
differential geometry.
As an example, one can cite the following result.
Consider the smooth stack B^∇G of smooth principal G-bundles
with connection and the smooth set Ω of differential forms.
The set of maps B^∇G→Ω turns out to be canonically
isomorphic to the algebra of Ad-invariant polynomials
on the Lie algebra of G.
Thus one recovers Chern-Weil theory in a very natural way.
See the recent paper by Freed and Hopkins for details:
http://arxiv.org/abs/1301.5959.
I don't think this result can be obtained in any other
model of smooth objects, because other models
do not allow for spaces like B^∇G and Ω.
A: For many applications, Banach manifolds are not suitable:
Groups of Sobolev or $C^k$ diffeomorphisms are only topological groups.
If a Banach Lie group acts effectively  on a compact (thus finite dimensional) smooth manifold, then it is finite dimensional itself.  
A: Different notions of manifolds may be useful in different approaches. Maybe more important than finding "universal consensus" on which one is suppoosed to be used where is to have a language to treat the various notions uniformly such as to be able pass between them in a useful way. 
One such more general category is that of "diffeological spaces". In 
http://ncatlab.org/nlab/show/diffeological+space
is discussed how for instance Frechet manifolds faithfully embed into these. 
Diffeological spaces form a "quasi-topos".  Following Grothendieck's lead, it is better to go one step further to an actual topos for differential geometry. The topos generalization of diffeological spaces is that of "smooth spaces" (smooth sets/smooth 0-types)
http://ncatlab.org/nlab/show/smooth+spaces
which is the sheaf topos over the category of smooth manifolds (or equivalently just over that of Euclidean spaces with smooth maps between them). Variants of this with a bit more information about the differential aspect of differential geometry include for instance the "Cahier topos"
http://ncatlab.org/nlab/show/Cahiers+topos
See there for pointers for how "convenient vector spaces" and hence the infinite-dimensional manifolds modeled on them ("convenient manifolds") are faithfully embedded into that topos.
In these toposes for instance all mapping spaces exist and can be usefully treated, while they agree with the infinite-dimensional manifold structures on mapping spaces whenever those actually exist. Similar statements hold for all other universal constructions.
Thereby topos theory transforms the question of finding "universal consensus" on which definition is best to a more relevant technical question: which concrete definition happens to constitute a presentation of a universaly existing construction in the topos. Presentations are useful, but are man-made. They may apply or not, may be useful here or there. But the smooth spaces which they present exist universally, robustly and meaniningfully irrespective of such choices.
