The possibility that the vacuum energy density (VED) could be a running quantity in the expanding Universe is intuitively more reasonable than just a rigid cosmological constant for the entire cosmic history. In fact, it may be more than a mere intuition. In the context of the running vacuum model (RVM), one finds that the VED evolves as a power series of the Hubble rate, H(t), and its time derivatives. This is not an ad hoc structure for it emerges from the result of detailed calculations in quantum field theory (QFT) in cosmological (FLRW) spacetime. The powers of H are just quantum effects from the vacuum fluctuations. As a result, the RVM i) predicts the running of both the cosmological `constant’ and the gravitational`constant’; ii) contributes to alleviate major issues such as the fine tuning conundrum within the cosmological constant problem; iii) furnishes a new mechanism for inflation; and iv) it implies that the (quantum) vacuum has a dynamical equation of state parameter (rather than the rigid value -1, as usually assumed), thus mimicking quintessence or phantom DE and potentially rendering the traditional DE fields expendable. Last, but not least, the RVM has a positive bearing on easing the current phenomenological tensions within the concordance model. In this talk, I will show that all of these properties are actually a generic prediction of QFT in FLRW spacetime. While this framework stems from old semi-qualitative ideas, only recently it has been possible to substantiate it after identifying an appropriate renormalization framework for tackling the issue of vacuum energy in the context of QFT and its connection with Cosmology.