Configuration schemes are a generalization of schemes which are obtained via formal gluing of the usual schemes along closed embeddings. In fact, the categories of quasicoherent sheaves are glued. Standard cohomology theory and the study of coherent sheaves extend to them. Configuration schemes were thought of to provide a nonclassical type of resolution of singularities, and they can be a part of a categorical version of resolution of singularities.
Valery A. Lunts, Coherent sheaves on configuration schemes, math.AG/0012061
Valery A. Lunts, Categorical resolution of singularities, math.AG/0905.4566