References on Donaldson Thomas theory I am aiming to start learning Donaldson-Thomas theory.  I have done the main parts of Vakil's notes and know some differential geometry.
Are there notes to give an introduction (or maybe more) on the subject? I am aware of the paper 13/2 ways to counting curves.  Would my background be enough to start reading this?  If my background is not enough, what do I lack and what do I need to bridge the gap?
 A: Let me begin with some thoughts about the prerequisites. Vakil's notes and Hartshorne are enough to start dealing with the theory. Additionally, you'll need to read about moduli theory and deformation theory. On deformation theory the first chapter of Hartshorne's book on the topic should be enough to begin with (just read it casually at first and come back to it as you deem it necessary). For moduli theory there are a lot of resources, I'll let you do the searching in here and mention that my first approaches to the theory were through the first chapter of Koch's book on GW theory and some lecture notes by Renzo Cavalieri (similar to these). 
Hilbet schemes are also very important for DT theory. Nakajima's book is a great reference, but you can also find some great notes online.
I'm pretty sure there's people in here who can give a more complete account on the literature, but I think I can give you some pointers. Perhaps most of what I'm saying is already mentioned in 13/2.
First of all, 13/2 is a great way to start since it describes the main features of all these enumerative theories and the relations between them. The references therein should be your first option here. I'd suggest you to try to learn a bit about GW, PT and GV too, since there's a lot of evidence and results showing there are deep connections between them. The presentation of GW, DT and PT described in there are correct, while the version of Gopakumar-Vafa they describe (the HST approach) is wrong in general (the newest proposal being that of Maulik and Toda). I don't know anything about the other approaches they discuss.
A milestone of the theory came with the MNOP papers, MNOP 1 and MNOP 2, where among other things the authors conjecture a very interesting relation between DT and GW theory of CY3s: they relate the partition functions on each side via the mysterious change of variables $-q\leftrightarrow e^{-i\lambda}$ and show it for the toric case. 
Finally, another great account of DT theory is this very nice paper by Balazs Szendroi, which is the only reference that doesn't appear in 13/2. 
