Over the past decades, the role of torsion in gravity has been extensivelyinvestigated along the main direction of bringing gravity closer to its gaugeformulation and incorporating spin in a geometric description. Here we reviewvarious torsional constructions, from teleparallel, to Einstein-Cartan, andmetric-affine gauge theories, resulting in extending torsional gravity in theparadigm of f(T) gravity, where f(T) is an arbitrary function of the torsionscalar. Based on this theory, we further review the corresponding cosmologicaland astrophysical applications. In particular, we study cosmological solutionsarising from f(T) gravity, both at the background and perturbation levels, indifferent eras along the cosmic expansion. The f(T) gravity construction canprovide a theoretical interpretation of the late-time universe acceleration,and it can easily accommodate with the regular thermal expanding historyincluding the radiation and cold dark matter dominated phases. Furthermore, ifone traces back to very early times, a sufficiently long period of inflationcan be achieved and hence can be investigated by cosmic microwave backgroundobservations, or alternatively, the Big Bang singularity can be avoided due tothe appearance of non-singular bounces. Various observational constraints,especially the bounds coming from the large-scale structure data in the case off(T) cosmology, as well as the behavior of gravitational waves, are describedin detail. Moreover, the spherically symmetric and black hole solutions of thetheory are reviewed. Additionally, we discuss various extensions of the f(T)paradigm. Finally, we consider the relation with other modified gravitationaltheories, such as those based on curvature, like f(R) gravity, trying toenlighten the subject of which formulation might be more suitable forquantization ventures and cosmological applications
