  We study left and right Bousfield localisations of stable model categories
which preserve stability. This follows the lead of the two key examples:
localisations of spectra with respect to a homology theory and A-torsion
modules over a ring R with A a perfect R-algebra. We exploit stability to see
that the resulting model structures are technically far better behaved than the
general case. We can give explicit sets of generating cofibrations, show that
these localisations preserve properness and give a complete characterisation of
when they preserve monoidal structures. We apply these results to obtain
convenient assumptions under which a stable model category is spectral. We then
use Morita theory to gain an insight into the nature of right localisation and
its homotopy category. We finish with a correspondence between left and right
localisation.
