  In this paper we are concerned with the problem of local and global
subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain
of a compact K\"ahler manifold. We prove that a precise bound on the complex
Monge-Amp\`ere mass of the given function implies the existence of a
subextension to a bigger regular subdomain or to the whole compact manifold. In
some cases we show that the maximal subextension has a well defined complex
Monge-Amp\`ere measure and obtain precise estimates on this measure. Finally we
give an example of a plurisubharmonic function with a well defined
Monge-Amp\`ere measure and the right bound on its Monge-Amp\`ere mass on the
unit ball in $\C^n$ for which the maximal subextension to the complex
projective space $\mb P_n$ does not have a globally well defined complex
Monge-Amp\`ere measure.
