The past decade has seen an explosion of progress in our understanding of scattering amplitudes in gauge theory and gravity. New bootstrap methods have revealed hidden symmetries and new mathematical structures that are completely invisible in the standard approach of Lagrangians and Feynman diagrams. Inspired by these developments, the bootstrap philosophy has recently been applied to cosmology. In this talk, I will describe our work on the bootstrapping of cosmological correlations. The talk will have two parts: In the first part, I will describe the conceptual foundations of the "cosmological bootstrap" as developed together with Arkani-Hamed, Lee and Pimentel in [arXiv:1811.00024]. In the second part, I will describe the extension of these ideas to massless particles with spin, where locality provides important new constraints. This is work to appear with Duaso Pueyo, Joyce, Lee and Pimentel. ----- Contact the organisers (Antoine Bourget and Edoardo Vescovi) for the link.