Anderson localization - an embarassment of riches I am looking for a good, not too technical discussion of Anderson Localization, and some explanation of why it exists. Googling "Anderson Localization" produces an infinite number  of possibilities, so perhaps someone knowledgeable can recommend something...
 A: There is also a new method which gets a hold of eigenvectors directly by an iterative diagonalization procedure rather than indirectly via expectations of products of resolvents.
It is in the recent article "Multi-Scale Jacobi Method for Anderson Localization" by John Imbrie.
I should add that the hot topic in the area now is many-body localization with tons of physics articles posted on the cond-mat section of arXiv. The above article by Imbrie can serve as an introduction to his other one about MBL: "On many-body localization for quantum spin chains" in JSP 2016. 
A: For a "canonical" list of references you might consult 50 Years of Anderson Localization. In addition to the Aizenman-Molchanov paper mentioned by Christian Remling,  the earlier Fröhlich-Spencer work was also quite influential.
A recent overview of the mathematics of Anderson localization is given by Günter Stolz:

We give a widely self-contained introduction to the mathematical
  theory of the Anderson model. After defining the Anderson model and
  determining its almost sure spectrum, we prove localization properties
  of the model. Here we discuss spectral as well as dynamical
  localization and provide proofs based on the fractional moments (or
  Aizenman-Molchanov) method. We also discuss, in less self-contained
  form, the extension of the fractional moment method to the continuum
  Anderson model. Finally, we mention major open problems.
We do not aim at the most general known results, but rather want to demonstrate that simple and natural mathematical ideas can be used to rigorously establish Anderson localization. 

A: In addition to the literature already recommended, maybe the following references will be also useful:
http://iopscience.iop.org/0034-4885/56/12/001 (Localization: theory and experiment, by B. Kramer and A. MacKinnon)
http://arxiv.org/abs/1005.0915 (Disorder and interference: localization phenomena, by C.A. Müller and D. Delande). 
A: Not technical, and not a complete answer, but illuminating:
https://www.quantamagazine.org/mathematicians-tame-rogue-waves-illuminating-future-of-led-lighting-20170822/

