It is likely that the early Universe was pervaded by a whole host of scalar fields which are ubiquitous in particle physics models and are employed everywhere from driving periods of accelerated expansion to the spontaneous breaking of gauge symmetries. Just as these scalar fields are important from a particle physics point of view, they can also have serious implications for the evolution of the Universe. In particular in extreme cases their dynamical evolution can lead to the failure of the synthesis of light elements or to exceed the dark matter bound in contrast to observation. These scalar fields are not however isolated systems and interact with the degrees of freedom which comprise their environment. As such two interrelated effects may arise; fluctuations and dissipation. These effects, which are enhanced at finite temperature, give rise to energy transfer between the scalar field and its environment and as such should be taken into account for a complete description of their dynamical evolution. In this thesis we will look at these effects within the inflationary era in a scenario termed warm inflation where amongst other effects, thermal fluctuations can now act as a source of primordial density perturbations. In particular we will show how a model of warm inflation based on a simple quartic potential can be brought back into agreement with Planck data through renormalizable interactions, whilst it is strongly disfavoured in the absence of such effects. Moving beyond inflation, we will consider the effect of fluctuation-dissipation dynamics on other cosmological scalar fields, deriving dissipation coefficients within common particle physics models. We also investigate how dissipation can affect cosmological phase transitions, potentially leading to late time periods of accelerated expansion, as well as presenting a novel model of dissipative leptogenesis.