  We present new definitions for and give a comprehensive treatment of the
canonical compactification of configuration spaces due to Fulton-MacPherson and
Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial
variant of this compactification initiated by Kontsevich. Our constructions are
elementary and give simple global coordinates for the compactified
configuration space of a general manifold embedded in Euclidean space. We
stratify the canonical compactification, identifying the diffeomorphism types
of the strata in terms of spaces of configurations in the tangent bundle, and
give completely explicit local coordinates around the strata as needed to
define a manifold with corners. We analyze the quotient map from the canonical
to the simplicial compactification, showing it is a homotopy equivalence. We
define projection maps and diagonal maps, which for the simplicial variant
satisfy cosimplicial identities.
