Cosmology of the early universe is a fascinating journey towards the origin of time. In addition, it offers a unique opportunity to investigate the fundamental laws of nature in regimes which are out of reach for particle physics experiments. I give a critical overview of the most popular paradigm, which is cosmological inflation. I review both the observational consequences of inflation and the technical and conceptual conundrums it leads to. I emphasize that inflation is an extremely UV-sensitive mechanism and hence it calls for an embedding in a UV-complete theory. I review some attempts and successes in building string theory models of inflation. I describe two case studies: brane inflation in a warped throat and axion monodromy inflation. I discuss conceptual advantages and shortcomings of these models. I conclude with some prospects for their detectability.
