Determining the nature of dark matter (DM) remains one of the key challenges in both particle physics and cosmology. Although we know the approximate distribution of DM in the Universe, we lack an understanding of its fundamental properties such as its mass and potential couplings to Standard Model particles. In the weakly-interacting massive particle (WIMP) paradigm, DM was in thermal equilibrium in the early Universe and we should expect scattering to have occurred between DM and Standard Model particles. In this thesis, we first consider the impact of primordial scattering between DM and radiation (photons or neutrinos). Such interactions give rise to a modification in the amplitude and position of the cosmic microwave background (CMB) acoustic peaks and a series of damped oscillations in the matter power spectrum. We obtain constraints from the Planck satellite and other CMB experiments, and then derive limits from large-scale structure (LSS) surveys. By providing forecasts for future experiments, we illustrate the power of LSS surveys in probing deviations from the standard cold DM (CDM) model. Then, using high-resolution N-body simulations, we show that the suppressed matter power spectra in such interacting DM scenarios allows one to alleviate the small-scale challenges faced by CDM; in particular, the "missing satellite" and "too big to fail" problems. Finally, we show that the excess of 511 keV gamma-rays from the Galactic centre, which has been observed by numerous experiments for four decades, cannot be explained via annihilations of light WIMPs, suggesting an astrophysical or more exotic DM source of the signal
