In this talk I will present you some of the results obtained during my Thesis, in which I have dealt with cosmological large-scale structure formation in the context of models alternative to the standard LCDM one, focusing in particular on possible modifications of General Relativity. \par In the first part of the talk I will introduce two modified-gravity theories, namely the K-mouflage and ultra-local models, and I will discuss their impact on observables, such as the galaxy power spectrum and the galaxy clusters scaling relations, and how this can be used to constrain these modified gravity models. \par In the second part I will focus on some non-perturbative results, the so-called ``consistency relations'', i.e. relations among correlators of cosmological fields, showing how these relations are a very promising tool to test General Relativity at scales usually hardly accessible to most of the current tests.