  The cosmological term, $\Lambda$, in Einstein's equations is an essential
ingredient of the `concordance' $\Lambda$CDM model of cosmology. In this
mini-review presentation, we assess the possibility that $\Lambda$ can be a
dynamical quantity, more specifically a `running quantity' in quantum field
theory in curved spacetime. A great deal of phenomenological works have shown
in the last few years that this option (sometimes accompanied with a running
gravitational coupling) may cure some of the tensions afflicting the
$\Lambda$CDM. The `running vacuum models' (RVM's) are characterized by the
vacuum energy density, $\rho_{\rm vac}$, being a series of (even) powers of the
Hubble rate and its time derivatives. Here we describe the technical quantum
field theoretical origin of the RVM structure in FLRW spacetime, which goes
well-beyond the original semi-qualitative renormalization group arguments. In
particular, we compute the renormalized energy-momentum tensor using the
adiabatic regularization procedure and show that it leads to the RVM form. In
other words, we find that the renormalized vacuum energy density,
$\rho_{vac}(H)$ evolves as a (constant) additive term plus a leading dynamical
components ${\cal O}(H^2)$. There are also ${\cal O}(H^4)$ contributions, which
can be relevant for the early universe. Remarkably enough, the renormalized
$\rho_{\rm vac}(H)$ does not exhibit dangerous terms proportional to the
quartic power of the masses ($\sim m^4$) of the fields. It is well-known that
these terms have been the main source of trouble since they are responsible for
the extreme fine tuning and ultimately for the cosmological constant problem.
In this context, the current $\rho_{vac}(H)$ is dominated by a constant term,
as it should be, but it acquires a mild dynamical component $\sim \nu H^2$
($|\nu|\ll1$) which makes the RVM to mimic quintessence.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory