We introduce the problem of quantum gravity, its motivations and challenges at the conceptual as well as physical level. In particular, we emphasize how constructing a new consistent quantum theory of the gravitational field implies deepening our understanding of the nature of space and time themselves. We overview existing theoretical approaches and proposals, and some recent results. Among the most recent suggestions, we illustrate the idea that spacetime itself could be understood as an emergent notion, and that geometry and topology may find their root in quantum entanglement. Finally (if time allows), we illustrate some of the general points by discussing one specific quantum gravity formalism, so-called group field theories, a promising recent blend (and evolution) of matrix/tensor models, loop quantum gravity and lattice quantum gravity, and how effective cosmological equations can be extracted from it.
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