  Wonderful compactifications of adjoint reductive groups over an algebraically
closed field play an important role in algebraic geometry and representation
theory. In this paper, we construct an equivariant compactification for adjoint
reductive groups over arbitrary base schemes. Our compactifications
parameterize classical wonderful compactifications of De Concini and Procesi as
geometric fibers. Our construction is based on a variant of the Artin-Weil
method of birational group laws. In particular, our construction gives a new
intrinsic construction of wonderful compactifications. The Picard group scheme
of our compactifications is computed. We also discuss several applications of
our compactification in the study of torsors under reductive group schemes.
