  Experiments with $\Xi^-$ atoms are proposed in order to study the nuclear
interaction of $\Xi$ hyperons. The production of $\Xi^-$ in the ($K^{-}$,
$K^{+}$) reaction, the $\Xi^-$ stopping in matter, and its atomic cascade are
incorporated within a realistic evaluation of the results expected for $\Xi^-$
X-ray spectra across the periodic table, using an assumed $\Xi$-nucleus optical
potential $V_{opt}$. Several optimal targets for measuring the
strong-interaction shift and width of the X-ray transition to the `last' atomic
level observed are singled out: F, Cl, I, Pb. The sensitivity of these
observables to the parameters of $V_{opt}$ is considered. The relevance of such
experiments is discussed in the context of strangeness -2 nuclear physics and
multistrange nuclear matter. Finally, with particular reference to searches for
the $H$ dibaryon, the properties of $\Xi^-d$ atoms are also discussed. The role
of Stark mixing, its effect on $S$ and $P$ state capture of $\Xi^-$ by the
deuteron, together with estimates of the resulting probability for producing
the $H$ dibaryon are considered in detail.
