BD-$\Lambda$CDM and Running Vacuum Models: Theoretical background and
  current observational status
  We present an analysis of the Brans-Dicke cosmological model with a
cosmological constant and cold dark matter (BD-$\Lambda$CDM). We find that the
BD-$\Lambda$CDM is favored by the overall cosmological data
(SNIa+BAO+$H(z)$+LSS+CMB) when it is compared with the standard model of
cosmology. The BD-$\Lambda$CDM model can be viewed from the GR perspective as a
Running Vacuum Model (RVM) with a time evolving vacuum energy density. Due to
this fact and also to its time evolving effective gravitational coupling, the
model can alleviate the $\sigma_8$ and the $H_0$ tensions at a time. We also
present the results for different types of RVM's when they are tested in the
light of the cosmological data and we show that a mild dynamics for the vacuum
energy density can help to smooth out the aforementioned tensions, thus
improving the performance of $\Lambda$CDM model.
