  The decays $B_d\to\pi^\mp K^\pm$ and $B^\pm\to\pi^\pm K$ provide interesting
constraints on the angle $\gamma$ of the unitarity triangle of the CKM matrix.
It is shown that bounds on $\gamma$ can also be obtained from the time
evolution of untagged $B_s\to K^+K^-$ and $B_s\to K^0\bar{K^0}$ decays,
provided the $B_s$ system exhibits a sizeable width difference
$\Delta\Gamma_s$. A detailed discussion of rescattering processes and
electroweak penguin effects, which limit the theoretical accuracy of these
constraints, and of methods to control them through experimental data is given.
Moreover, strategies are pointed out to go beyond these bounds by relating the
$B_{u,d}\to\pi K$ and untagged $B_s\to K\bar{K}$ decays through the SU(3)
flavour symmetry of strong interactions. If a tagged, time-dependent
measurement of the $B_s\to K^+K^-$ and $B_s\to K^0\bar{K^0}$ modes should
become possible, $\gamma$ could be determined from the corresponding
observables in a way that makes use of only the SU(2) isospin symmetry and
takes into account rescattering effects ``automatically''. The impact of
new-physics contributions to $B^0_s - \bar{B^0_s}$ mixing is also analysed, and
interesting features arising in such a scenario of physics beyond the Standard
Model are pointed out.
