  We investigate chaotic inflation models with two scalar fields, such that one
field (the inflaton) rolls while the other is trapped in a false vacuum state.
The false vacuum becomes unstable when the inflaton field falls below some
critical value, and a first or second order transition to the true vacuum
ensues. Particular attention is paid to Linde's second-order `Hybrid
Inflation'; with the false vacuum dominating, inflation differs from the usual
true vacuum case both in its cosmology and in its relation to particle physics.
The spectral index of the adiabatic density perturbation can be very close to
1, or it can be around ten percent higher. The energy scale at the end of
inflation can be anywhere between $10^{16}$\,GeV and $10^{11}$\,GeV, though
reheating is prompt so the reheat temperature can't be far below
$10^{11}\,$GeV. Topological defects are almost inevitably produced at the end
of inflation, and if the inflationary energy scale is near its upper limit they
can have significant effects.
  Because false vacuum inflation occurs with the inflaton field far below the
Planck scale, it is easier to implement in the context of supergravity than
standard chaotic inflation. That the inflaton mass is small compared with the
inflationary Hubble parameter is still a problem for generic supergravity
theories, but remarkably this can be avoided in a natural way for a class of
supergravity models which follow from orbifold compactification of
superstrings. This opens up the prospect of a truly realistic, superstring
