Vacuum dynamics in the Universe versus a rigid $\Lambda=$const

In this year, in which we celebrate 100 years of the cosmological term, $\Lambda$, in Einstein's gravitational field equations, we are still facing the crucial question whether $\Lambda$ is truly a fundamental constant or a mildly evolving dynamical variable. After many theoretical attempts to understand the meaning of $\Lambda$, and in view of the enhanced accuracy of the cosmological observations, it seems now mandatory that this issue should be first settled empirically before further theoretical speculations on its ultimate nature. In this work, we summarize the situation of some of these studies. Devoted analyses made recently show that the $\Lambda=$const. hypothesis, despite being the simplest, may well not be the most favored one. The overall fit to the cosmological observables $SNIa+BAO+H(z)+LSS+CMB$ singles out the class RVM of the "running" vacuum models, in which $\Lambda=\Lambda(H)$ is an affine power-law function of the Hubble rate. It turns out that the performance of the RVM as compared to the "concordance" $\Lambda$CDM model (with $\Lambda=$const.) is much better. The evidence in support of the RVM may reach $\sim 4\sigma$ c.l., and is bolstered with Akaike and Bayesian criteria providing strong evidence in favor of the RVM option. We also address the implications of this framework on the tension between the CMB and local measurements of the current Hubble parameter.