CP Violation in the $B_s^0$ system

We review experimental and theoretical aspects of CP violation in the $B_s^0$-system. In that respect we present updates for the mixing parameters of the $B_s^0$ mesons expected in the Standard Model (SM): the mass difference $\Delta M_s^{\rm SM} = (18.3 \pm 2.7)$ ps$^{-1}$, the decay rate difference $\Delta \Gamma_s^{\rm SM} = (0.085 \pm 0.015)$ ps$^{-1}$, the flavour specific CP asymmetry $a_{\rm fs}^{s, \rm SM} = (2.22 \pm 0.27) \cdot 10^{-5}$ and the equivalent quantities in the $B^0$-sector. The experimental measurements of $\Delta M_s$ and $\Delta \Gamma_s$ are more precise and agree very well with theoretica predictions, which can also be viewed as a proof of theoretical tools, like the Heavy Quark Expansion (HQE). CP violating studies in the $B_s^0$ system provide essential information to test the SM expectations. Mixing quantities can be used to study model independent bounds on new physics (NP) effects, yielding the familiar picture: the standard model (SM) gives the dominant contribution, while there is still some sizable space for NP effects. In the case of $\Delta M_s$ effects of the order of $15\%$ are not excluded. In the CP phase $\phi_s$ due to CP violation in the interference of decays and mixing ${\cal O} (100 \%)$ effects are still possible. The semi-leptonic CP asymmetry $a_{\rm sl}^s$, which is due to CP violation in mixing, could still be a factor of 250 larger than its robust SM expectation and thus provides a very clean observable for NP searches. We comment in detail on theoretical improvements that are necessary to make full use of the experimental precision.