  This paper reviews the investigations on the vacuum expectation value of the
current density for a charged scalar field in spacetimes with toroidally
compactified spatial dimensions. As background geometries locally Minkowskian
(LM), locally de Sitter (LdS) and locally anti-de Sitter (LAdS) spacetimes are
considered. Along compact dimensions quasiperiodicity conditions are imposed on
the field operator and the presence of a constant gauge field is assumed. The
vacuum current has non-zero components only along compact dimensions. Those
components are periodic functions of the magnetic flux enclosed by compact
dimensions with the period equal to the flux quantum. For LdS and LAdS
geometries and for small values of the length of a compact dimension, compared
with the curvature radius, the leading term in the expansion of the the vacuum
current along that dimension coincides with that for LM bulk. In this limit the
dominant contribution to the mode sum for the current density comes from the
vacuum fluctuations with wavelength smaller than the curvature radius and the
influence of the gravitational field is weak. The effects of the gravitational
field are essential for lengths of compact dimensions larger than the curvature
radius. In particular, instead of the exponential suppression of the current
density in LM bulk one can have power law decay in LdS and LAdS spacetimes.
