In this thesis, we study the infrared behaviour of propagators in Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes, of which de Sitter is a special case. For the most part, we are interested in the infrared behaviour of the graviton two-point function, in FLRW spacetime. Naively, it is thought that the two-point function requires an infrared cut-off in order to be well-defined. However, we find a gauge transformation such that the two-point function can be rendered IR finite, for a large class of FLRW spacetimes. The graviton two-point function also experiences an infrared divergence when the separation between the two points is taken to be large. In de Sitter spacetime, in the Landau gauge, this divergence is found to be logarithmic. However, it is found that this logarithmic divergence can also be removed by means of a suitable non-covariant gauge transformation. In finding the large-distance behaviour of the graviton two-point function, we initially found it useful to find the large-distance behaviour of the covariant massless vector propagator, in de Sitter spacetime. Through this calculation, we were able to find a method which could be extended to the more computationally complex case of the graviton two-point function.