We make a universal construction of Bruhat-Tits group scheme on wonderful embeddings of adjoint groups in the absolute and relative settings of adjoint Kac-Moody groups. These group schemes have natural classifying properties reflecting the orbit structure on the wonderful embeddings. A theme which is reflected in the work is that the various wonderful embeddings give algebra-geometric realizations of the different kinds of buildings. This shows an intimate relationship between the wonderful embeddings and Bruhat-Tits theory.