Some extensions of the Standard Model provide Dark Matter candidate particles with sub-GeV mass. These Light Dark Matter particles have been considered for example in Warm Dark Matter scenarios (e.g. the keV scale sterile neutrino, axino or gravitino). Moreover MeV scale DM candidates have been proposed in supersymmetric models and as source of the 511 keV line from the Galactic center. In this paper the possibility of direct detection of a Light Dark Matter candidate is investigated considering the inelastic scattering processes on the electron or on the nucleus targets. Some theoretical arguments are developed and related phenomenological aspects are discussed. Allowed volumes and regions for the characteristic phenomenological parameters of the considered scenarios are derived from the DAMA/NaI annual modulation data.