The scientific understanding of the Universe is evolving at a rapid pace. Each new experiment yields more and more accurate measurements of its fundamental parameters. The standard cosmological model postulates a very early period of fast expansion, but the details of its underlying mechanism remain hidden behind a veil of high energies that we cannot access in particle accelerators. Physics of the early Universe can instead be studied by identifying its effects on cosmological fluctuations produced in the early Universe, which are responsible for the anisotropies of the Cosmic Microwave Background and the development of cosmic structure we can observe today. On the other hand, General Relativity, in its classical formulation, is not fully compatible with the principles of quantum mechanics, and a theory connecting the two realms remains to be discovered. Direct experimental verification of such a theory is challenging due to the extremely high energies required. Therefore, cosmological perturbations provide an excellent window into the perturbative regime of quantum gravity. Since the primordial perturbations were produced in the highly energetic early Universe, they can in principle be used to distinguish between different quantum gravity models. It is therefore essential to develop methods of deriving their statistics from specific features of the models. This thesis focuses on the cosmological bootstrap, a research program that attempts to derive features of cosmological fluctuations from simple physical principles expected to be satisfied in the early Universe. I study the effect of background curvature on standard soft theorems and its impact on observables in the context of the Effective Field Theory of inflation. I extend flat spacetime bootstrap methods to settings where the boost symmetry is violated. I also employ several well-known cosmological bootstrap methods to constrain graviton correlators at the end of inflation.