  We perform a general study of the issue of metastability for
supersymmetry-breaking vacua in theories with N=1 and N=2 global supersymmetry.
This problem turns out to capture all the important qualitative features of the
corresponding question in theories with local supersymmetry, where
gravitational effects induce only quantitative modifications. Moreover, it
allows to directly compare the conditions arising in the N=1 and N=2 cases,
since the latter becomes particular case of the former in the rigid limit. Our
strategy consists in a systematic investigation of the danger of instability
coming from the sGoldstini scalars, whose masses are entirely due to
supersymmetry breaking mass-splitting effects. We start by reviewing the
metastability conditions arising in general N=1 non-linear sigma-models with
chiral and vector multiplets. We then turn to the case of general N=2
non-linear sigma-models with hyper and vector multiplets. We first reproduce
and clarify the known no-go theorems applying to theories with only Abelian
vector multiplets and only hyper multiplets, and then derive new results
applying to more general cases. To make the comparison with N=1 models as clear
as possible, we rely on a formulation of N=2 models where one of the
supersymmetries is manifestly realized in terms of ordinary superfields,
whereas the other is realized through non-trivial transformations. We give a
self-contained account of such a construction of N=2 theories in N=1
superspace, generalizing previous work on various aspects to reach a general
and coordinate-covariant construction. We also present a direct computation of
the supertrace of the mass matrix.
