  Despite the outstanding achievements of modern cosmology, the classical
dispute on the precise value of $H_0$, which is the first ever parameter of
modern cosmology and one of the prime parameters in the field, still goes on
and on after over half a century of measurements. Recently the dispute came to
the spotlight with renewed strength owing to the significant tension (at
$>3\sigma$ c.l.) between the latest Planck determination obtained from the CMB
anisotropies and the local (distance ladder) measurement from the Hubble Space
Telescope (HST), based on Cepheids. In this work, we investigate the impact of
the running vacuum model (RVM) and related models on such a controversy. For
the RVM, the vacuum energy density $\rho_{\Lambda}$ carries a mild dependence
on the cosmic expansion rate, i.e. $\rho_{\Lambda}(H)$, which allows to
ameliorate the fit quality to the overall $SNIa+BAO+H(z)+LSS+CMB$ cosmological
observations as compared to the concordance $\Lambda$CDM model. By letting the
RVM to deviate from the vacuum option, the equation of state $w=-1$ continues
to be favored by the overall fit. Vacuum dynamics also predicts the following:
i) the CMB range of values for $H_0$ is more favored than the local ones, and
ii) smaller values for $\sigma_8(0)$. As a result, a better account for the LSS
structure formation data is achieved as compared to the $\Lambda$CDM, which is
based on a rigid (i.e. non-dynamical) $\Lambda$ term.
