  In this survey we give a brief introduction to, and review the progress made
in the last decade in understanding the geometry of the moduli spaces A_g of
principally polarized abelian varieties and its compactifications. Topics
surveyed include: compactifications; birational geometry: nef and effective
cones, canonical models; homology, Chow rings and intersection theory; and
subvarieties of moduli spaces. We also discuss some open problems and possible
further directions. This is an expanded and updated version of the talk given
at the 2005 Summer Institute for Algebraic Geometry
