Cosmological Evolution of the Universe in Torsion-based Modified Gravity

General Relativity, despite its century-long success, faces conceptual and observational challenges, including singularities, incompatibility with quantum mechanics, and the need to introduce dark matter and dark energy. Precision cosmology has also revealed persistent tensions, notably the H0 and S8 discrepancies, which question the completeness of the standard Lambda-CDM model. This thesis investigates cosmological applications of teleparallel gravity and its extensions, focusing on f(T) and f(T,T) theories. We show that torsion-based modifications can shift late-time expansion and matter clustering, alleviating the H0 and S8 tensions. Using datasets including cosmic chronometers, baryon acoustic oscillations, Type Ia supernovae, Pantheon+SH0ES, Union3, DESI, and gravitational wave standard sirens, we perform Markov Chain Monte Carlo analyses to constrain model parameters. Model-independent diagnostics using cosmography demonstrate that extended teleparallel theories can be tightly constrained. Pade approximations and direct dynamical reconstructions yield consistent results, with some models outperforming Lambda-CDM using recent DESI and Union3 data. We also propose a connection between late-time acceleration and early-Universe baryogenesis, showing that torsional gravity can reproduce the observed baryon asymmetry while remaining consistent with late-time expansion. Overall, teleparallel gravity provides a robust alternative framework capable of alleviating key cosmological tensions and linking early- and late-Universe physics.