  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. These notes
are based on several lecture series which the author gave at the Kochi School
on Random Schr\"odinger Operators, November 26-28, 2009, the Arizona School of
Analysis and Applications, March 15-19, 2010 and the Summer School on
Mathematical Physics, Sogang University, July 20-23, 2010.
