  Current astronomical observations are successfully explained by the present
cosmological paradigm based on the concordance model ($\Lambda_0$CDM +
Inflation). However, such a scenario is composed of a heterogeneous mix of
ingredients for describing the different stages of cosmological evolution.
Particularly, it does not give an unified explanation connecting the early and
late time accelerating inflationary regimes which are separated by many aeons.
Other challenges to the concordance model include: a singularity at early times
or the emergence of the Universe from the quantum gravity regime, the
"graceful" exit from inflation to the standard radiation phase, as well as, the
coincidence and cosmological constant problems. We show here that a simple
running vacuum model or a time-dependent vacuum may provide insight to some of
the above open questions (including a complete cosmic history), and also can
explain the observed matter-antimatter asymmetry just after the initial
deflationary period.
