TITLE: Roadmap to study Atiyah–Singer index theorem
QUESTION [51 upvotes]: I am a physics undergrad and want to pursue a PhD in maths (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non departmental courses, thought I will be able to take them is a couple of semesters. Anyway, So I was thinking about taking a mini-project of sorts which I can self-study  or maybe ask a prof. I had heard about this theorem while studying topological solitons in physics. 
What are the prerequisites for studying the Atiyah–Singer theorem. I had a look online, but I couldn't figure out exactly, in what field this is, or what are the prerequisites. Wikipedia says it is a theorem in differential geometry, but obviously what differential geometry I know is insufficient. I know about manifolds, differential forms, lie derivatives, lie groups, and some killing vectors stuff. A look at the material online also talks about some operators in DG which I have never come across. Are there an algebraic topology prerequisites?
Please could you recommend me some references which can take me there? 
Thanks in advance!

REPLY [12 votes]: There is a new wonderful, big and deep book dealing precisely with everything you need (assuming knowledge of "basic" differential geometry):

Bleecker; Booß-Bavnek - Index Theory with Applications to Mathematics and Physics, International Press 2013, (792 pages!).

These are the same authors of the following old classic, but the new book is NOT a new edition but a completely NEW title (also with modern TeX typeset, figures and additional topics and details):

Bleecker; Booß-Bavnek - Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics, Springer Verlag 1985.

You may also find in this other answer a very brief overview of the proof of Atiyah-Singer and why pseudo-differential operators are needed. There you will find links to lectures and other titles regarding the theorem.