  The determination of the mixing induced CP-violating asymmetry in decays such
as $B^0_s \to J/\psi \phi$ 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. The steps towards a
precise determination of this phase with a flavour-tagged, time-dependent,
angular analysis of the decay $B^0_s \to J/\psi \phi$ will be presented, and
first results shown from this measurement programme, using data collected in
2010 and the early months of the 2011 run. Results will also be shown, and
prospects discussed, from related measurements.
