In this thesis, we have studied gauged and geometric vector fields at the MeV scale in two main parts. The basic framework of these two parts are given briefly as follows. In the first part (Chapter \ref{chapter-U(1)}), we have built a family-nonuniversal $U(1)^\prime$ model populated by an MeV-scale sector with a minimal new field content which explains the recent anomalous beryllium decays. Excited beryllium has been observed to decay into electron-positron pairs with a $6.8~\sigma$ anomaly. The process is properly explained by a $17$ MeV proto-phobic vector boson. In this thesis, we consider a family-nonuniversal $U(1)^{\prime}$ that is populated by the $U(1)^{\prime}$ gauge boson $Z^\prime$ and a scalar field $S$. The kinetic mixing of $Z^\prime$ with the hypercharge gauge boson, as we show by a detailed analysis, generates the observed beryllium anomaly. We show that beryllium anomaly can be explained by an MeV-scale sector with a minimal new field content. In the second part (Chapter \ref{chapter-GDM}), we have shown how a light vector particle can arise from metric-affine gravity and how this particle fits the current data and constraints on the dark matter. We show that, metric-affine gravity , which involves metric tensor and affine connection as two independent fields, dynamically reduces, in its minimal form, to the usual gravity plus a massive vector field. The vector $Y_\mu$ is neutral and long-living when its mass range lies in the range $9.4\ {\rm MeV} < M_Y < 28.4\ {\rm MeV}$. Its scattering cross section from nucleons, which is some 60 orders of magnitude below the current bounds, is too small to facilitate direct detection of the dark matter. This property provides an explanation for whys and hows of dark matter searches. We show that due to its geometrical origin the $Y_\mu$ couples only to fermions. This very feature of the $Y_\mu$ makes it fundamentally different than all the other vector dark matter candidates in the literature. The geometrical dark matter we present is minimal and self-consistent not only theoretically but also astrophysically in that its feebly interacting nature is all that is needed for its longevity.