  Despite the many outstanding cosmological observations leading to a strong
evidence for a nonvanishing cosmological constant (CC) term in the
gravitational field equations, the theoretical status of this quantity seems to
be lagging well behind the observational successes. It thus seems timely to
revisit some fundamental aspects of the CC term in Quantum Field Theory (QFT).
We emphasize that, in curved space-time, nothing a priori prevents this term
from potentially having a mild running behavior associated to quantum effects.
Remarkably, this could be the very origin of the dynamical nature of the Dark
Energy, in contrast to many other popular options considered in the literature.
In discussing this possibility, we also address some recent criticisms
concerning the possibility of such running. Our conclusion is that, while there
is no comprehensive proof of the CC running, there is no proof of the
non-running either. The problem can be solved only through a deeper
understanding of the vacuum contributions of massive quantum fields on a curved
spacetime background. We suggest that such investigations are at the heart of
one of the most important endeavors of fundamental theoretical cosmology in the
years to come.
