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- The recent acceleration of the UniversePublication . Albarran Payo, Imanol; Bouhmadi Lopez, Mariam; Marto, João Pedro de JesusThe present thesis is aimed to disclose three genuine phantom Dark Energy (DE) models where each of them induce a particular cosmic doomsday. We have named these models as model A, model B and model C, while the corresponding induced cosmic events are known as Big Rip (BR), Little Rip (LR) and Little Sibling of the Big Rip (LS), respectively. We regard a BR as a true singularity since it takes place at a finite cosmic time, while we have coined LR and LS as abrupt events, since they occur at infinite cosmic time. Nevertheless, it is well known that in such abrupt events sooner or later all the bound structures would unavoidably torn away, and therefore, the Universe would face a total destruction at a finite cosmic time. On the one side, we have addressed the background phenomenology and the first order cosmological perturbations for the phantom DE models above mentioned. In addition, we have made use of the widely known ΛCDM model as a guideline to measure deviations among the models. Given that a DE content is present, we avoid the associated instabilities at the perturbative level by applying the method of DE pressure decomposition in its adiabatic and non‐adiabatic contributions. We compute, by means of numerical methods, the evolution of the perturbed quantities for a Universe filled with radiation, matter and DE. Such computations are carried from well inside the radiation dominated era to the far future. Then, we predict the current matter power spectrum and fσ8 growth rate for each model. The latter mentioned observable quantity is compared with the current observational data in order to find footprints that could allow us to distinguish between the mentioned models. For the sake of completeness, we have fitted observationally these phantom DE models together with ΛCDM in order to constrain the parameters characterising the models. On the one hand, we have found that despite that ΛCDM still gives the best fit, it is closely followed by the models studied in the present thesis. On the other hand, we have found that these genuine phantom models induce a sign switch of the gravitational potential at very large scale factors. This fact could be understood as gravity becoming effectively repulsive in the far future. Finally, we have studied the effects of DE speed of sound on the perturbations. On the other side, it is expected that quantum effects will become important when the Universe approaches a future cosmic singularity, which is the case of those events addressed in the present thesis. Unfortunately, we have not yet a consistent theory of quantum gravity to deal with the most dramatic effects that would take place at the end of the Universe. It is expected that such a fundamental quantum theory of gravity will naturally avoid those singularities present in the classical theory of General Relativity (GR). We have rather addressed the issue of cosmological singularity avoidance within the context of a quantum approach. The quantisation is carried via Wheeler‐DeWitt (WDW) equation and imposing the DeWitt (DW) boundary condition, i.e. the wave function vanishes close to the singularity. We have analysed each model by considering different factor orderings and solving the WDW equation for a DE content given by (i), a perfect fluid, and (ii), a scalar field. In addition, we have addressed these phantom models in the context of the Eddington‐inspired‐Born‐Infeld (EiBI) modified theory of gravity and applied the same quantisation methods above mentioned to analyse the avoidance of singularities from a quantum point of view. Therefore, this thesis is divided in two main parts, a classical part, where we present the back-ground and perturbations of three genuine phantom models, and a second part, where we address the avoidance of singularities induced by such models from a quantum point of view. Given that UBI allows to present the thesis as an introduction, a set of chapters based on the published works during the PhD and the conclusions, we have followed mainly this format.