"Measurement of the muon anomalous magnetic moment

to high and even higher precision"

                             David  Hertzog


I will discuss the recent series of measurements

of the muon anomalous magnetic moment (g-2), which

were carried out at the Brookhaven National Laboratory.

The final result---representing the average of three

running periods with positive and negative muons---has

a relative uncertainty of 0.5 ppm.  Comparison to

theory reveals a slight deviation from the standard

model expectation if the e+e- based hadronic vacuum

polarization evaluation is used.  Further improvements

in the theory are in progress.  What's next?  Our

g-2 Collaboration has recently submitted a new proposal

to Brookhaven with the aim to reduce the experimental

uncertainty to 0.2 ppm.  The techniques being developed

and the challenges that lie ahead to achieve this step

forward will be discussed in this talk.