My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me get started with the basics? I've seen Lang's book but that's it.

2$\begingroup$ For infinite dimensional smooth and Riemannian manifolds, Serge Lang's books are popular as a start. $\endgroup$ – Claudio Gorodski Mar 30 '14 at 2:30

4$\begingroup$ Why don't you list (in the question) the books you've already "seen", then? $\endgroup$ – Francois Ziegler Mar 30 '14 at 8:35

3$\begingroup$ Klingenberg's Riemannian geometry was recommended reading when I was an undergrad, and uses manifolds modeled on Banach spaces. It might help to say a bit more about what you need to learn. $\endgroup$ – S. Carnahan♦ Mar 30 '14 at 21:53

2$\begingroup$ Andrew Stacey's 2007 talk "Variations on a Theme: Riemannian Geometry in Infinite Dimensions" is eminently digestible: only 7 pages. PDF download $\endgroup$ – Joseph O'Rourke Mar 30 '14 at 23:18

1$\begingroup$ I've been alerted to this question. The right thing to read depends on the context. What infinite dimensional spaces are you interested in? Specifically, what model spaces for your spaces? (Please edit your question with that information.) $\endgroup$ – Loop Space Apr 2 '14 at 16:56
Lempert, László The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces. Invent. Math. 142 (2000), no. 3, 579603.
Lempert, László The Dolbeault complex in infinite dimensions. II. J. Amer. Math. Soc. 12 (1999), no. 3, 775793.
Lempert, László The Dolbeault complex in infinite dimensions. I. J. Amer. Math. Soc. 11 (1998), no. 3, 485520.
Hamilton, Richard S. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.

$\begingroup$ Thanks! A colleague just turned me on to the Hamilton paper, it seems to be a good entry point for the subject. $\endgroup$ – Wintermute Apr 3 '14 at 11:23
I would also suggest the beautiful paper by Arnold
http://www.ams.org/mathscinetgetitem?mr=202082
and the book by
KrieglMichor: The convenient setting of global analysis,
see also the references therein.