By directly probing the initial conditions of our universe, cosmological surveys offer us a unique observational handle on quantum field theory in curved spacetime with dynamical gravity and might even allow us to glean information about a full theory of quantum gravity. Here we report on recent progress to study the natural observables in the problem, namely cosmological correlators. After setting the stage, we review results from three different approaches. First, we present the in-out formalism as an interesting alternative to the well-known in-in formalism and stress some of its advantages, such as the derivation of recursion relations, correlators cutting rules and a proposal for a de Sitter scattering matrix. Second, we tackle the important open problem of constructing effective theories in curved spacetime, which generally requires an open quantum system approach. Third, we provide an executive summary of general properties of the field-theoretic wavefunction that follow from symmetries, unitarity, causality and locality. We describe how these properties can be leveraged to bootstrap all tree-level results and we discuss loop contributions.