The paper reviews various torsional constructions in gravity, including teleparallel, Einstein-Cartan, and metric-affine gauge theories, leading to the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. The authors discuss the cosmological and astrophysical applications of f(T) gravity, focusing on cosmological solutions at different stages of cosmic expansion, the late-time universe acceleration, inflation, and cosmic bounce. They also explore observational constraints, such as those from large-scale structure data and gravitational waves, and review spherical symmetric and black hole solutions. Additionally, the paper examines extensions of the f(T) paradigm, including non-minimally coupled scalar-torsion gravity, f(T,T_G) gravity, and f(R,T) teleparallel gravity. The authors compare torsion and curvature gravity and provide a summary of the main findings.The paper reviews various torsional constructions in gravity, including teleparallel, Einstein-Cartan, and metric-affine gauge theories, leading to the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. The authors discuss the cosmological and astrophysical applications of f(T) gravity, focusing on cosmological solutions at different stages of cosmic expansion, the late-time universe acceleration, inflation, and cosmic bounce. They also explore observational constraints, such as those from large-scale structure data and gravitational waves, and review spherical symmetric and black hole solutions. Additionally, the paper examines extensions of the f(T) paradigm, including non-minimally coupled scalar-torsion gravity, f(T,T_G) gravity, and f(R,T) teleparallel gravity. The authors compare torsion and curvature gravity and provide a summary of the main findings.