Based on many astrophysical and cosmological observations, it is clear that the Universe contains some form of non-relativistic, non-interacting form of matter called Dark Matter (DM). The Standard Model of particle physics (SM) does not have a particle with all the suitable properties to explain all of these observations. We look to model the DM through different extensions of the SM ensuring that one or more new particles have all the desired properties. We study bounds on these models through theoretical constraints such as perturbative unitarity and stability of the scalar potential, as well as through experimental constraints such as direct and indirect detection. Of particular interest is the measurement of the relic density which allows us to make a direct connection between the parameters of our models and the amount of DM in the Universe today through a process known as freeze-out. We look at different freeze-out scenarios such as single component and multi-component DM, as well as alternative cosmological scenarios. We find that the parameter space for single component DM is severely constrained, while multi-component DM can escape bounds quite easily because of the many freedoms in the models. We find that changes in the cosmology in the early Universe can significantly change the requirements of a DM model, leading to much more weakly coupled or heavier DM.