Neutrino oscillations constitute the first experimental evidence of new physics. Con- trary to the prediction of the Standard Model, neutrinos are now well-established as particles with mass. In this thesis we study four topics related to neutrino masses. We examine a class of patterns of Majorana neutrino mass matrices, called two-zero textures, that could lead to the building of the underlying flavour theory. Using a purely numerical technique, we compare the constraints given by the textures with the most updated data of oscillation parameters. We find that the most promising textures are those of class A with normal ordering. Further, we revisit the case of excluded textures and study the stability of the results when the textures are only approximate. Then, we present a new model of neutrino masses that is able to generate an A class texture. Due to the form of the mass matrix, the standard neutrino exchange mechanism is suppressed; however, a neutrinoless double beta decay rate large enough to be tested in the near future is induced through a short range mechanism. Moreover, this model belongs to a broad variety of frameworks that are based on the interaction of a lepton number violating charged scalar with light leptons. The signatures of such scalars, often accompanied by missing energy, are not currently being probed at high-energy accelerators. We propose a search strategy and examine the prospects for their signatures at the Large Hadron Col- lider. Finally, we consider the possibility that neutrinos are Dirac particles. In this context, right-handed neutrinos need to be properly included in the effective field theory of the Standard Model. Some of the resulting effective operators would show interesting phenomenological impact that can be addressed with current experimen- tal data. Furthermore, our results would indicate how further investigations should be designed in the future in order to test the unbounded operators to shed light on possible new physics. Accordingly, we propose a new search in the context of rare top decays.