Review on f(Q) gravity

Recent years have witnessed a rise in interest in the geometrical trinity of General Relativity and its extensions. This interest has been fuelled by novel insights into the nature of gravity, the possibility to address computational and conceptual questions—such as the determination of black hole entropy or the definition of gravitational energy–momentum—from a new perspective. In particular, $f\left(Q\right)$ gravity has also inspired numerous works on black holes, wormholes, and cosmology. In the latter case, $f\left(Q\right)$ models have the potential to elucidate phenomena in both early and late-time cosmology without necessitating the inclusion of dark energy, the inflaton field, or dark matter. Particularly noteworthy is the role of $f\left(Q\right)$ theories in addressing cosmological tensions, presenting exciting possibilities for reshaping our understanding of gravity and its manifestations in cosmology. The emergence of intriguing new black hole solutions and the potential existence of wormhole solutions suggest the presence of novel physics within the realm of strong gravity. These phenomena have become increasingly measurable only in recent times, opening up exciting avenues for further exploration and discovery. This review is tailored to students and researchers alike. It offers a self-contained and pedagogical introduction to metric-affine geometry—The mathematical foundation and indispensable tool upon which the geometrical trinity of General Relativity as well as its various extensions are built.