A multitude of evidence has accumulated in support of the existence of dark matter in our Universe. There are already plenty of dark matter candidates. However, we do not know yet whether any of these candidates constitutes the whole or a part of the dark matter population despite the tremendous experimental efforts. In this thesis, we study several possible dark matter production mechanisms and the corresponding observational and theoretical constraints in the context of inflationary cosmology. Adopting a model-independent approach, we explore the parameter space for dark matter with a mass of order MeV and above showing that only small regions of the parameter space for the popular freeze-out mechanism are still viable. Nevertheless, the regions of the parameter space corresponding to the freeze-in and non-thermal dark matter scenarios are mostly unexplored. We, therefore, zoom into these regions and show that a connection to the inflationary observables can be established, which can help constrain these scenarios. We then consider the parameter space of a sub-eV dark matter candidate, the axion. We show that using the Cosmic Microwave Background radiation constraint on the effective number of relativistic species, an interesting constraint can be placed. This bound arises from the fact that the field whose angular excitations are the axions can be displaced from its minimum during inflation and later decays dominantly into ultra-relativistic, axions which contribute to the effective number of relativistic species. We finally consider the possible production of axion-like particle via non-perturbative effects due to their coupling to inflatons or moduli. We show that this mechanism is efficient only if the amplitude of inflaton/moduli oscillations is initially much larger than the mass scale associated with the axion-like particles. In this case, bounds can be placed on the corresponding parameter spaces.