  Using standard quantum field theory, we discuss several theoretical aspects
of the possible running of the cosmological constant (CC) term in Einstein's
equations. The basic motivation for the present work is to emphasize that this
possibility should also be taken into account when considering dynamical models
for the dark energy (DE), which are nowadays mainly focused on identifying the
DE with the energy density associated to one or more ad hoc scalar fields. At
the same time, we address some recent criticisms that have been published (or
privately communicated to us) attempting to cast doubts on the fundamental
possibility of such running. In this work, we argue that while there is no
comprehensive proof of the CC running, there is no rigorous proof of the
non-running either. In particular, some purported "non-running theorem"
recently adduced in the literature is, in our opinion, completely insubstantial
and formally incorrect. The way to the CC running is, therefore, still open and
we take here the opportunity to present a pedagogical review of the present
state of the art in this field, including a a brief historical account.
