By Bernstein center one usually means the center of the category of
representations of a reductive p-adic groups.
I will review some basic definitions and results related to it - in
particular, I will explain the so called "spectral" and "geometric"
descriptions of the Bernstein center. After that I will talk about
some recent results (due to several people such as Bezrukavnikov,
Kazhdan, Lusztig, Varshavsky and myself) related to the (rather
mysterious) interplay between the above descriptions of the Bernstein
center.