  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 vacuum
models (RVM's), which can be motivated in the context of Quantum Field Theory
in curved spacetime. They consider the possibility of a smoothly evolving
vacuum energy density that inherits its time-dependence from cosmological
variables, such as the Hubble rate and its time derivative. The analysis
presented, however, is not limited only to the dynamical vacuum models but
considers also models of dark energy such as the Peebles \& Ratra scalar field
model or simple parameterizations like the XCDM or the CPL. Their theoretical
predictions are tested against the wealth of cosmological data and a fitting
procedure is carried out in order to obtain constraints over the free
parameters that characterize the models under study. Finally, an extended
discussion about the $\sigma_8$ and $H_0$ tensions is provided and a possible
solution for both of them is described within the framework of Brans and Dicke
gravity with a cosmological constant, which mimics the RVM with a mild
time-evolving gravitational coupling $G$. The main conclusion of this study is
that appreciable signals in favour of a time-evolving dark energy density are
found in the current data.
