  Perhaps the deepest mystery of our accelerating Universe in expansion is the
existence of a tiny and rigid cosmological constant, $\Lambda$. Its size is
many orders of magnitude below the expected one in the standard model of
particle physics. However, an expanding Universe is not expected to have a
static vacuum energy density. We should rather observe a mildly dynamical
behavior $\delta\Lambda(t)\sim R\sim H^2(t)$ with the expansion rate $H$. At
the same time, it is natural to think that the huge value of the primeval
vacuum energy (presumably connected to some grand unified theory) was
responsible for the initial inflationary phase. In the traditional inflaton
models such phase is inserted by hand in the early epoch of the cosmic
evolution, and it is assumed to match the concordance $\Lambda$CDM regime
during the radiation epoch. Here, instead, we consider a class of dynamical
vacuum models which incorporate into a single vacuum structure
$\bar{\Lambda}(H)$ the rapid stage of inflation, followed by the radiation and
cold matter epochs, until achieving our dark energy Universe. The early
behavior of the model bares resemblance with Starobinsky's inflation and
ptovides a solution to the large entropy problem. It is compatible with the
latest cosmological data on Hubble expansion and structure formation, and
presents distinctive observational features that can be tested in the near
future.
