  We present the constraints on the parameters of several light boson mediator
models obtained from the analysis of the current data of the COHERENT CE$\nu$NS
experiment. We consider a variety of vector boson mediator models: the
so-called universal, the $B-L$ and other anomaly-free $U(1)'$ gauge models with
direct couplings of the new vector boson with neutrinos and quarks, and the
anomaly-free $L_e-L_\mu$, $L_e-L_\tau$, and $L_\mu-L_\tau$ gauge models where
the coupling of the new vector boson with the quarks is generated by kinetic
mixing with the photon at the one-loop level. We consider also a model with a
new light scalar boson mediator that is assumed, for simplicity, to have
universal coupling with quarks and leptons. Since the COHERENT CE$\nu$NS data
are well-fitted with the cross section predicted by the Standard Model, the
analysis of the data yields constraints for the mass and coupling of the new
boson mediator that depend on the charges of quarks and neutrinos in each model
under consideration. We compare these constraints with the limits obtained in
other experiments and with the values that can explain the muon $g-2$ anomaly
in the models where the muon couples to the new boson mediator.
