  We discuss two types of instabilities which may arise in string theory
compactified to asymptotically AdS spaces: perturbative, due to discrete modes
in the spectrum of the Laplacian, and non-perturbative, due to brane
nucleation. In the case of three dimensional Einstein manifolds, we completely
characterize the presence of these instabilities, and in higher dimensions we
provide a partial classification. The analysis may be viewed as an extension of
the Breitenlohner-Freedman bound. One interesting result is that, apart from a
very special class of exceptions, all Euclidean asymptotically AdS spaces with
more than one conformal boundary component are unstable, if the
compactification admits BPS branes or scalars saturating the
Breitenlohner-Freedman bound. As examples, we analyze quotients of AdS in any
dimension and AdS Taub-NUT spaces, and show a space which was previously
discussed in the context of AdS/CFT is unstable both perturbatively and
non-perturbatively.
