  We study in an effective operator approach how the effects of new physics
from various scenarios containing extra $Z'$ gauge bosons or doubly charged
scalars can affect, and thus be tested by the precision polarized Moeller
scattering experiment. We give Wilson coefficients for various classes of
models, and we deduce constraints on the parameter space of the relevant
coupling constants and mixing angles from the results of the SLAC E158
experiment, and that projected from the future high precision JLAB experiment.
