The talk starts with an elementary introduction to the Cosmological Constant
(CC) problem. There are actually several distinct problems, one of them is
that many people apply the cutoff regularization in a very naive way and
misinterpret the results as a fundamental difficulty. From the other side
there is a real and very difficult CC problem, which results from the existence
of many independent contributions to the same small quantity, that is the CC,
or, equivalently, energy density of vacuum. The real CC problem consists in
the enormous precision of the requested finetuning for one of these
contributions. The CC problem can be seen as mainly a Quantum Field Theory
hierarchytype problem, which comes from the "conflict" between the particle
physics scale where we evaluate the mentioned contributions and the tiny cosmic
scale where we observe their sum. The development and improvement of the
methods of Quantum Field Theory in curved spacetime is necessary for better
understanding of the CC problem, in both early and late universe cases. In
particular, there is a possibility of variable CC due to the quantum effects
of matter fields, which can be described by the renormalization group running
of CC in the far IR. Since our presentday knowledge of the Appelquist and
Carazzone decoupling theorem in gravity is restricted, the possibility of
such a running can not be theoretically proved or ruled out at the presentday
state of art in the field, and this opens the door for some interesting
phenomenological applications in cosmology and astrophysics.
