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Of the four known fundamental interactions, gravity remains, to date, the most mysterious one. In fact, the quest for a consistent theory of quantum gravity is believed to be deeply intertwined with some of the most outstanding open questions in high energy physics and cosmology. In this colloquium we will discuss some recent proposals for universal constraints which every effective field theory must obey if it arises as a low-energy limit of a consistent quantum gravity theory. As a first circle of ideas, we introduce the Emergent String Conjecture, which puts severe constraints on the asymptotic weak coupling regime of a gravitational effective theory. As strong evidence in its favour, we show how this principle is deeply imprinted in the classical and quantum geometry of non-perturbative string and M-theory compactifications, and, as suggested by ongoing work, on positivity in the effective theory. We also discuss some of its most important general implications, which include the well-known Weak Gravity Conjecture as a corollary in the asymptotic regime and which recently have also lead to more speculative ideas related to particle physics and cosmology. Time permitting, we furthermore exemplify how probe arguments can put novel bounds on the particle content of consistent gravity theories and on the positivity of some of its higher couplings.