  We establish a general gluing theorem for constant mean curvature solutions
of the vacuum Einstein constraint equations. This allows one to take connected
sums of solutions or to glue a handle (wormhole) onto any given solution. Away
from this handle region, the initial data sets we produce can be made as close
as desired to the original initial data sets. These constructions can be made
either when the initial manifold is compact or asymptotically Euclidean or
asymptotically hyperbolic, with suitable corresponding conditions on the
extrinsic curvature. In the compact setting a mild nondegeneracy condition is
required. In the final section of the paper, we list a number ways this
construction may be used to produce new types of vacuum spacetimes.
