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Dark matter direct detection experiments still have found no evidence of the dark matter WIMPs. The search has therefore been expanded for lighter dark matter candidates. Light dark matter is nearly invisible to current detectors through the elastic nuclear recoils. This thesis is meant to provide understanding on the inelastic atomic scatterings, which are one good way to detect dark matter particles with mχ ∼ GeV. In this thesis we consider spin-independent scatterings. Inelastic scatterings are based on the fact that in an atom, electrons do not follow the motion of the recoil nucleus immediately, but instead it takes time. This results in a small probability of observable ionization or excitation of the atom. This is known as the Migdal effect. We will first study the theoretical framework of dark matter-nucleus scatterings, showing how to get the event rate and how it is factorized into the astrophysical, the particle physics and the target response part. Then we will move to the inelastic processes, Migdal and Bremsstrahlung effects, deriving their event rates. In the first, we try to detect ionized electrons. The latter one, the Bremsstrahlung, is a similar process to the Migdal, but there we try to detect photons emitted from the de-excitations of atoms excited in the inelastic recoils. We will also look into the Migdal in semiconductors. Because of the smaller gap for electron excitations in crystals, we find that the rate for the Migdal effect is much higher in semiconductors than in atomic targets, thus allowing the search for even lighter dark matter particles. The rate can be expressed in terms of the energy loss function of the target material