  The stability of our vacuum is analyzed and several aspects concerning this
question are reviewed. 1) In the standard Glashow-Weinberg-Salam (GWS) model we
review the instability towards the formation of a bubble of lower energy
density and how the rate of such bubble formation process compares with the age
of the Universe for the known values of the GWS model. 2) We also review the
recent work by one of us (E.I.G) concerning the vacuum instability question in
the context of a model that solves the cosmological constant problem. It turns
out that in such model the same physics that solves the cosmological constant
problem makes the vacuum stable. 3) We review our recent work concerning the
instability of elementary particle embedded in our vacuum, towards the
formation of an infinite Universe. Such process is not catastrophic. It leads
to a "bifurcation type" instability in which our Universe is not eaten by a
bubble (instead a baby universe is born). This universe does not replace our
Universe rather it disconnects from it (via a wormhole) after formation.
