  Recently a variety of nonlocal integrable systems has been introduced that
besides fields located at particlar space-time points simultaneously also
contain fields that are located at different, but symmetrically related,
points. Here we investigate different types of soliton solutions with regard to
their stability against linear pertubations obtained for the nonlocal version
of the Hirota/nonlinear Schr\"odinger equation and the so-called Alice and Bob
versions of the Korteweg-de Vries and Bousinesq equations. We encounter
different types of scenarios: Solition solutions that are linearly stable or
unstable and also solutions that change their stability properties depending on
the parameter regime they are in.
