This paper reviews the theory of teleparallel gravity and its applications in cosmology. It discusses the role of torsion in gravity, starting from the general theory of relativity (GR), which is based on the curvature of spacetime. The paper explores various torsional constructions, including teleparallel, Einstein-Cartan, and metric-affine gauge theories, leading to the extension of torsional gravity in the paradigm of $ f(T) $ gravity, where $ f(T) $ is an arbitrary function of the torsion scalar. The paper reviews the corresponding cosmological and astrophysical applications of $ f(T) $ gravity, including cosmological solutions in different eras of cosmic expansion, the late-time universe acceleration, inflation, cosmic bounce, and observational constraints. It also discusses the behavior of gravitational waves, black hole solutions, and various extensions of $ f(T) $ gravity, such as non-minimally coupled scalar-torsion gravity, $ f(T, T_G) $ gravity, and $ f(R, T) $ teleparallel gravity. The paper compares torsion and curvature gravity and concludes with a summary of the key findings and implications for quantization and cosmological applications.This paper reviews the theory of teleparallel gravity and its applications in cosmology. It discusses the role of torsion in gravity, starting from the general theory of relativity (GR), which is based on the curvature of spacetime. The paper explores various torsional constructions, including teleparallel, Einstein-Cartan, and metric-affine gauge theories, leading to the extension of torsional gravity in the paradigm of $ f(T) $ gravity, where $ f(T) $ is an arbitrary function of the torsion scalar. The paper reviews the corresponding cosmological and astrophysical applications of $ f(T) $ gravity, including cosmological solutions in different eras of cosmic expansion, the late-time universe acceleration, inflation, cosmic bounce, and observational constraints. It also discusses the behavior of gravitational waves, black hole solutions, and various extensions of $ f(T) $ gravity, such as non-minimally coupled scalar-torsion gravity, $ f(T, T_G) $ gravity, and $ f(R, T) $ teleparallel gravity. The paper compares torsion and curvature gravity and concludes with a summary of the key findings and implications for quantization and cosmological applications.