  In this paper we introduce the characteristic gluing problem for the Einstein
vacuum equations. We present a codimension-$10$ gluing construction for
characteristic initial data which are close to the Minkowski data and we show
that the $10$-dimensional obstruction space consists of gauge-invariant charges
which are conserved by the linearized null constraint equations. By relating
these $10$ charges to the ADM energy, linear momentum, angular momentum and the
center-of-mass we prove that asymptotically flat data can be characteristically
glued (including the $10$ charges) to the data of a suitably chosen Kerr
spacetime, obtaining as a corollary an alternative proof of the Corvino--Schoen
spacelike gluing construction. Moreover, we derive a localized version of our
construction where the given data restricted on an angular sector is
characteristically glued to the Minkowski data restricted on another angular
sector. As a corollary we obtain an alternative proof of the Carlotto-Schoen
localized spacelike gluing construction. Our method yields no loss of decay in
the transition region, resolving an open problem. We also discuss a number of
other applications.
