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In this PhD thesis a wide variety of cosmological models beyond the$\Lambda$CDM are studied in detail. Great emphasis is put on the running vacuummodels (RVM's), which can be motivated in the context of Quantum Field Theoryin curved spacetime. They consider the possibility of a smoothly evolvingvacuum energy density that inherits its time-dependence from cosmologicalvariables, such as the Hubble rate and its time derivative. The analysispresented, however, is not limited only to the dynamical vacuum models butconsiders also models of dark energy such as the Peebles \& Ratra scalar fieldmodel or simple parameterizations like the XCDM or the CPL. Their theoreticalpredictions are tested against the wealth of cosmological data and a fittingprocedure is carried out in order to obtain constraints over the freeparameters that characterize the models under study. Finally, an extendeddiscussion about the $\sigma_8$ and $H_0$ tensions is provided and a possiblesolution for both of them is described within the framework of Brans and Dickegravity with a cosmological constant, which mimics the RVM with a mildtime-evolving gravitational coupling $G$. The main conclusion of this study isthat appreciable signals in favour of a time-evolving dark energy density arefound in the current data.

Implications of Dynamical Dark Energy in the expansion of the Universe and the Structure Formation