This dissertation is an investigation of the conceptual and physical foundations of cosmological inflation. Cosmologists have long claimed that inflation solves certain fine-tuning problems with the previous standard model of cosmology, the highly successful big bang model of the universe. The major contributions of the dissertation are an analysis and a critique of the arguments cosmologists could be construed as making to support this claim, arguments which have been taken as the standard rationale for the initial (and even continuing) widespread acceptance of inflation in cosmology. There are four chapters. In the first I establish the context of the investigation by clarifying the notion of a cosmological model (the central component of which is a relativistic spacetime) and explaining important features of the cosmological models underlying the big bang model. In the second chapter I analyze and criticize the aforementioned standard motivation for the inclusion of inflation in modern cosmological models and address the question of whether inflation solves the big bang model's fine-tuning problems. The main conclusion of this chapter is that there is at present no good argument that inflation solves the big bang model's fine-tuning problems. In chapters three and four I delve deeper into the fine-tuning argument apparently favored by cosmologists, namely the one that depends on interpreting fine-tuning in terms of probabilities or likelihoods. In chapter three I show how probabilities are implemented, interpreted, and justified in classical statistical physics, introducing a novel interpretation of statistical mechanics along the way. In the final chapter I put the formal implementations, the interpretations, and the justification of probability from chapter three to work in the context of cosmology. I ultimately claim that the many challenges I raise in the chapter are decisive for current and even forseeable approaches to defining cosmological probabilities.