  In this thesis we present detailed analyses on various running vacuum models
(RVM's), in which the vacuum energy density "runs" with the cosmic expansion.
The RVM's are motivated from the renormalization group formalism of Quantum
Field Theory in curved spacetime. In this theoretical framework, the vacuum
energy density depends explicitly on the Hubble function and its time
derivative and thus does not remain constant, in contradistinction to the
standard $\Lambda$CDM picture. Upon studying how these models fit the data, we
can determine whether this dynamical behavior is favored by observations or
not. Is the vacuum energy density (or more generically, the DE density)
dynamical? One of the main conclusions of this dissertation is that we have
indeed strong evidences in favor of the variability of the DE density
throughout the cosmic history. It is shown that this variation can also be
traced through other (purely phenomenological) dynamical vacuum models,
together with different parametrizations of the DE (as the XCDM and CPL), and
scalar field models as the Peebles-Ratra one. The statistical confidence level
of these evidences reaches in some cases the 3-4$\sigma$ c.l., something that
is unprecedented in the literature. An exhaustive explanation of the reasons
why large collaborations as BOSS, Planck and DES have not been able to detect
such signal in favor of the DE dynamics is provided in this thesis too.
