I will review our current understanding of the initial conditions of the universe, and describe what information is available from current and future measurements of cosmological correlation functions. Then I will describe a new method to compute and constrain the possible shapes of those correlation functions, assuming they were generated during inflation. This ``cosmological bootstrap” draws inspiration from the modern scattering amplitudes program in flat space, as well as the conformal bootstrap of phase transitions. After discussing primordial scalar fluctuations, I will also explain how the consistent propagation of gravitational waves imposes further constraints on the structure of spinning primordial correlators.