  We describe the closure of the strata of abelian differentials with
prescribed type of zeros and poles, in the projectivized Hodge bundle over the
Deligne-Mumford moduli space of stable curves with marked points. We provide an
explicit characterization of pointed stable differentials in the boundary of
the closure, both a complex analytic proof and a flat geometric proof for
smoothing the boundary differentials, and numerous examples. The main new
ingredient in our description is a global residue condition arising from a full
order on the dual graph of a stable curve.
