  This is a survey paper based on my talks at the 41st Symposium on Ring Theory
and Representation Theory, held in Shizuoka University, Japan in September
2008, and will appear in the conference proceedings.
  The paper begins with a brief introduction to the use of Hochschild
cohomology in developing the theory of support varieties for a module over an
artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004),
705-732). The paper then describes the current status of research concerning
the structure of the Hochschild cohomology ring modulo nilpotence.
