Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this talk, we will construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes, which 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. If time permits, we will also discuss several applications of our compactification in the study of torsors under reductive group schemes.