  Neutral gauge bosons beyond the Standard Model are becoming interesting as
possible mediators to explain several experimental anomalies. They have small
masses, below one GeV, and are referred to as dark photons, $U$, $A'$ or $Z'$
bosons. Electron--positron collision experiments at the B-factories provide the
most straightforward way to probe bosons of this kind. In the present article
we study production of the bosons at $e^+e^-$ colliders operating at GeV
center-of-mass energies. We have studied two channels: $e^+e^-\rightarrow
\gamma Z'$ and $e^+e^-\rightarrow e^+e^-Z'$. Analytic expressions for the cross
sections and various observables such as the energy spectra of the produced
bosons and the final electrons from the $Z'$ decays are derived. We have also
studied the transverse momentum distribution of the bosons and the spatial
distribution of the $Z'\rightarrow e^+e^-$ decay vertices. It is shown that
these distributions provide distinct signatures of the bosons in
$e^+e^-\rightarrow\gamma Z'$. The reaction $e^+e^-\rightarrow e^+e^-Z'$ becomes
important at small $Z'$ scattering angles where its contribution to the overall
yield may be larger by orders of magnitude compared to $e^+e^-\rightarrow\gamma
Z'$. The standard processes $e^+e^-\rightarrow\gamma\gamma$ and
$e^+e^-\rightarrow e^+e^-\gamma$ that lead to the same signal are considered.
We include numerical predictions for the production rates at the energy
$\sqrt{s}=10.5$ GeV. The case with a light scalar boson is also discussed. The
calculations are performed in detail and can be useful for additional studies.
