We briefly review the torsional construction of gravitation dubbed as f(T) gravity, where f is an arbitrary function of the torsion scalar. We review its cosmological applications in different eras of the cosmic history. This construction can provide an interpretation of the late-time acceleration, alternative to a cosmological constant. If one traces back to very early times, inflation can be achieved for a class of extended torsional gravity models. Moreover, the big bang singularity can be avoided in the model of f(T) bounce. A discussion on observational constraints of f(T) gravity was presented focusing on the realization of quintom scenario. This proceeding article is based on an accompanied review~1 and a talk given in the the parallel session "Emergent Gravity and Quantum Gravity" in the 2nd LeCosPA International Symposium - Everything about Gravity.
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