We analyse present constraints on the SM parameter space and derive, in a model independent way, various bounds on New Physics contributions to B 0 d -B 0 d and B 0 s -B 0 s mixings. Our analyses include information on a large set of asymmetries, leading to the measurement of the CKM phases γ andβ, as well as recent data from D0 and CDF related to the B 0 s -B 0 s system such as the measurement of ∆M Bs , A and ∆Γ CP s . We examine in detail several observables such as the asymmetries A d sl , A, the width differences ∆Γ d and ∆Γ CP s and discuss the rôle 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 phaseχ entering in B 0 s -B 0 s mixing and study the impact of a future measurement ofχ to be achieved at LHC, through the measurement of the time-dependent CP asymmetry in B s → J/Ψ Φ decays.