At low energies, the world around us can be described very accurately using the Standard Model and \Lambda CDM. However, these are at best only ``effective'' descriptions: valid at low energies but destined to break down as experiments in particle physics and cosmology probe increasingly higher energies, ultimately requiring a new (UV complete) theory to take over. In this talk, I will review the constraints which must be placed on our low-energy Effective Field Theories if they are to have any hope of a smooth Wilsonian UV completion at high energies (which is unitary, causal and local). These constraints are known as ``positivity bounds''. Exploiting such connections between the IR and the UV in order to constrain our IR model-building is particularly important for cosmology and gravity, as they are able to rule out large regions of parameter space on purely theoretical grounds (i.e. only certain models could ever make physical sense at high energies), complementing and strengthening our ever-improving experimental tests. For instance, by applying positivity constraints as theoretical priors on a Monte Carlo Markov Chain analysis, we have recently improved the experimental constraints on Horndeski scalar-tensor models by a factor of 100. And this is just using our current CMB, BAO, matter power spectrum and RSD data (Planck/SDSS/BOSS/6dF)---as our future cosmological measurements increase in precision, the role of positivity in determining good theoretical priors will play a crucial part in strengthening our data analysis, improving our observational constraints, and guiding our theories of gravity.