This thesis describes a detailed investigation of the effects of matter inhomogeneities on the cosmological evolution of the fine structure constant using the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) theory. We briefly review the observational and theoretical motivations to this work, together with the standard cosmological model. We start by analysing the phase space of the system of equations that describes a time-varying fine structure constant, in a homogeneous and isotropic background universe. We classify all the possible behaviours of the fine structure constant in ever-expanding universes and find exact solutions for it. Using a gauge-invariant formalism, we derive and solve the linearly perturbed Einstein cosmological equations for the BSBM theory. We calculate the time evolution of inhomogeneous perturbations of the fine structure constant on small and large scales with respect to the Hubble radius. We also investigate, in the non-linear regime of the large scale structure formation, the space-time evolution of the fine structure constant, inside evolving spherical overdensities. The dependence on the dark-energy equation of state is also analysed. Finally, we analyse the effects of the coupling of the field (that drives the variations in the fine structure constant) to the matter fields, on the space and time evolution of the fine structure constant.
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