We review the progress made in the determination of the weak charge, Q\_w, of
the cesium nucleus which raises the status of Atomic Parity Violation
measurements to that of a precision electroweak test. Not only is it necessary
to have a precision measurement of the electroweak asymmetry in the highly
forbidden 6S-7S transition, but one also needs a precise calibration procedure.
The 1999 precision measurement by the Boulder group implied a 2.5 sigma
deviation of Q\_w from the theoretical prediction. This triggered many particle
physicist suggestions as well as examination by atomic theoretical physicists
of several sources of corrections. After about three years the disagreement was
removed without appealing to "New Physics". Concurrently, an original
experimental approach was developed in our group for more than a decade. It is
based on detection by stimulated emission with amplification of the left- right
asymmetry. We present our decisive, recent progress together with our latest
results. We emphasize the important impact for electroweak theory, of future
measurements in cesium possibly pushed to the 0.1% level. Other possible
approaches are currently explored in several atoms