If dark matter, after it has become non-relativistic, scatters elastically with a relativistic heat bath particle, then the resulting pressure leads to acoustic oscillations that suppress the growth of overdensities in the dark matter fluid. If such an interaction can keep dark matter in kinetic equilibrium until keV temperatures, this effect then suppresses structure formation on scales roughly equal to dwarf galaxy scales and smaller, possibly addressing the missing satellite problem. The goal of this thesis is to study the possibilities for such late kinetic decoupling in particle models for dark matter. Using the Boltzmann equation, we discuss the thermal decoupling process of dark matter in detail. In addition to discussing specific dark matter models, we also go into important general considerations and requirements for late kinetic decoupling, and models with dark radiation. We summarize the results obtained in Bringmann et al., 2016, but go into more details on two specific models. First a model consisting of two real scalar particles, one dark matter particle, and one relativistic dark radiation particle, interacting through a 4-particle vertex. This model is of particular interest not only because it is so simple, but also because a large class of effective field theory models will also essentially map onto this model. When combining relic density constraints with late kinetic decoupling, we need very light dark matter (m < MeV). For these masses, the assumption that dark matter is highly non-relativistic during chemical decoupling breaks down. However, when the dust settles, we find that this is still a viable model for late kinetic decoupling. We also study a model where a fermionic dark matter particle transforms in the fundamental representation of some SU(N) gauge group. The scattering in the t-channel is so enhanced at low energies in this model, that kinetic decoupling does not happen until the dark radiation becomes non-relativistic. As we discuss, depending on what happens to the dark radiation temperature when it becomes non-relativistic, the resulting suppression of dark matter structures can be radically different. In any case these models seem to require a low value for the dark radiation temperature, which is hard to achieve in model building without new input.