Mixing and $CP$-violation in the $B_{s}^{0}$ system at LHCb

The determination of the $CP$-violating phase $\phi_s$ in $B_s^0\to J/\psi\phi$ decays is one of the key goals of the LHCb experiment. Its value is predicted to be very small in the Standard Model but can be significantly enhanced in many models of new physics. To perform the first LHCb analysis of $\phi_s$ on 2010 data at a centre-of-mass energy of 7 TeV, many milestones needed to be achieved first, such as the measurements of the $b$-hadron lifetimes, the optimization and calibration of the flavour-tagging algorithms, the measurement of the polarization amplitudes in $B^0\to J/\psi K^{*0}$ decays, of the $B_s^0$ mixing frequency $\Delta m_s$ and of the $CP$ asymmetry in $B^0\to J/\psi K_S^0$. We will present our result of the first $\phi_s$ analysis and related measurements. Additionally we will show signals of several $B_s^0$ decay modes that have been observed for the first time at the LHCb experiment and which can potentially be used to extract information on φs such as $B_s^0\to J/\psi f_0$ or $B_s^0\to K^{*0}\bar{K}^{*0}$