One of the longest-standing questions in modern cosmology concerns the mechanism that sets the initial conditions for the evolution of matter in the Universe. In particular, what is the physics underlying the initial inflationary epoch? Or, more radically, did inflation take place or not? To answer this question, one typically relies on observational data for fluctuations that are small enough to be described by standard cosmological perturbation theory. Nonetheless, data outside the perturbative regime are very abundant and precisely measured, but usually discarded because hard to model. It would be ideal to have a tool able to take advantage of this data. In light of this, I will discuss how to robustly extract information about the inflationary mechanism, and the non-Gaussianity it generates, using non-perturbative methods. I will start with an introduction to standard cosmological perturbation theory, with emphasis on its regime of applicability. Then, I will introduce the main tool: the consistency relations for large-scale structures. These are non-perturbative identities applicable to cosmological correlators. Finally, I will show how they can be used to extract the amplitude of local primordial non-Gaussianity from those non-linear data to which perturbation theory does not apply.