  We analyse present constraints on the SM parameter space and derive, in a
model independent way, various bounds on New Physics contributions to
$B_d^0$--$\bar B_d^0$ and $B_s^0$--$\bar B_s^0$ mixings. Our analyses include
information on a large set of asymmetries, leading to the measurement of the
CKM phases $\gamma$ and $\bar\beta$, as well as recent data from D0 and CDF
related to the $B_s^0$--$\bar B_s^0$ system such as the measurement of $\Delta
M_{B_s}$, $A_{SL}$ and $\Delta\Gamma_{s}^{CP}$. We examine in detail several
observables such as the asymmetries $A_{sl}^d$, $A_{SL}$, the width differences
$\Delta\Gamma_{d}$ and $\Delta\Gamma_{s}^{CP}$ and discuss the r\^ole they play
in establishing the limits on New Physics. The present data clearly favour the
SM, with the New Physics favoured region placed around the SM solution. A New
Physics solution significantly different from the SM is still allowed, albeit
quite disfavoured (2.6% probability). We analyse the presently available
indirect knowledge on the phase $\bar\chi$ entering in $B_s^0$--$\bar B_s^0$
mixing and study the impact of a future measurement of $\bar\chi$ to be
achieved at LHC, through the measurement of the time-dependent CP asymmetry in
$B_s\to J/\Psi \Phi$ decays.
