  In this thesis we are interested in understanding how Lorentz symmetry
violation can affect some features of fermion dynamics and, perhaps, help to
solve some well-known problems in particle physics, such as the origin of
neutrino masses and oscillations.
  Firstly, we consider two Lorentz-Invariance-Violating (LIV) models and
investigate the possibility of generating masses and oscillations dynamically
for both Dirac and Majorana neutrinos, using non-perturbative methods, such as
the Schwinger-Dyson and the effective potential approaches. In our studies,
Lorentz symmetric models are extended by the inclusion of higher-order LIV
operators, which improve the convergence of loop integrals and introduce a
natural mass scale to the theories. We then present how Lorentz invariance can
be recovered, for both models, after quantisation, in such a way that the
dynamical masses and mixing are the only quantum effects that remain finite.
  Additionally, we study how matter fields, especially fermions, behave when
coupled to two modified gravity models. Such modified gravity models break the
4-dimensional diffeomorphism invariance and, consequently, induce local Lorentz
violation. In particular, we consider Horava-Lifshitz gravity, which presents
an improved ultraviolet behaviour when compared to General Relativity (GR), and
thus addresses a fundamental problem in physics: the
non-(perturbative-)renormalisability of the theory of GR. We calculate the LIV
one-loop corrections to the matter sector dispersion relations, after
integration over graviton components, and show that, by imposing reasonable
constraints on the energy scales of our gravity models, our results are
consistent with the current bounds on Lorentz symmetry violation.
