TITLE: Theories of Noncommutative Geometry
QUESTION [44 upvotes]: [I have rewritten this post in a way which I hope will remain faithful to the questioner and make it seem more acceptable to the community.  I have also voted to reopen it. -- PLC]
There are many ways to approach noncommutative geometry. 
What are some of the most important currently known approaches?  Who are the principal creators of each of these approaches, and where are they coming from?  E.g., what more established mathematical fields are they using as jumping off points?  What problems are they trying to solve?
Two examples: 
1) The Connes school, with an approach from C$^*$-algebras/mathematical physics.
2) The Kontsevich school, with an approach from algebraic geometry.

REPLY [9 votes]: There is a beautiful three-part series of lectures by Jonathan Block that introduces both the Connes and Kontsevich schools of non-commutative geometry:
http://www.math.upenn.edu/~tpantev/rtg09bc/lecnotes/block-ncg.pdf
One of the main motivating examples throughout the lectures is Lusztig's proof proof of the Novikov conjecture.