  This article surveys some recent work of the author on Hilbert modular
fourfolds X. After some preliminaries on the cohomology and special,
codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate
conjecture for X over abelian fields is exposed. It is followed by a
description of a method to construct classes in the Bloch's Chow group
CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable
intersections in translates of modular curves. The article ends with the
introduction of a modular Gersten complex for a general Shimura variety X and
the corresponding groups CH^p_{mod}(X) and CH^p_{mod}(X,1).
