Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and
Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space,
whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K.
This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre
compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact,
they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of
Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these
compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H
in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding
into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms,
in particular Eisenstein series.