The Cosmological Constant Problem and Running Vacuum in the Expanding
  Universe
  It is well-known that quantum field theory (QFT) induces a huge value of the
cosmological constant, $\Lambda$, which is outrageously inconsistent with
cosmological observations. We review here some aspects of this fundamental
theoretical conundrum (`the cosmological constant problem') and strongly argue
in favor of the possibility that the cosmic vacuum density $\rho_{\rm vac}$ may
be mildly evolving with the expansion rate $H$. Such a `running vacuum model'
(RVM) proposal predicts an effective dynamical dark energy without postulating
new ad hoc fields (quintessence and the like). Using the method of adiabatic
renormalization within QFT in curved spacetime we find that $\rho_{\rm vac}(H)$
acquires a dynamical component ${\cal O}(H^2)$ caused by the quantum matter
effects. There are also ${\cal O}(H^n)$ ($n=4,6,..$) contributions, some of
which may trigger inflation in the early universe. Remarkably, the evolution of
the adiabatically renormalized $\rho_{\rm vac}(H)$ is not affected by dangerous
terms proportional to the quartic power of the masses ($\sim m^4$) of the
fields. Traditionally, these terms have been the main source of trouble as they
are responsible for the extreme fine tuning feature of the cosmological
constant problem. In the context under study, however, the late time $\rho_{\rm
vac}(H)$ around $H_0$ is given by a dominant term ($\rho_{\rm vac}^0$) plus the
aforementioned mild dynamical component $\propto \nu (H^2-H_0^2)$ (with
$|\nu|\ll1$), which makes the RVM to mimic quintessence. Finally, on the
phenomenological side we show that the RVM may be instrumental in alleviating
some of the most challenging problems (so-called `tensions') afflicting
nowadays the observational consistency of the `concordance' $\Lambda$CDM model,
such as the $H_0$ and $\sigma_8$ tensions.
