  This is the note for the four lectures given by the author in the
``International Short-School/Conference on Affine Algebraic Geometry and the
Jacobian Conjecture" at Chern Institute of Mathematics, Nankai University,
Tianjin, China. July 14-25, 2014.
  The aim of this lectures is to give an introduction to the theory of Mathieu
subspaces. We will not treat all topics in their most general setting, but will
restrict to certain classes of "nice" rings. As we shall see even for these
rings there is a lot we don't know yet!
