The talk includes a kind of a pedagogical introduction to the Cosmological Constant (CC) problem. The application of the cut-off regularization in a naive way leads to misinterpretation of the result as a fundamental difficulty, which in fact does not exist. From the other side there is a very difficult CC problem, which results from
the existence of many independent contributions to the same (very small) quantity, that is the density of the CC. The real CC problem consists in the enormous precision of the requested fine-tuning for one of these contributions. The CC problem is essentially the hierarchy 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. It is clear that the use of Quantum Field Theory methods in curved space-time is necessary for a 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. This possibility can not be proved or ruled out at the present-day state of art in the field, and this opens
the door for many interesting phenomenological applications in both cosmology and astrophysics.