  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.
