  We review recent works on the possibility for eternal existence of thin-shell
wormholes on Einstein and Einstein-Gauss-Bonnet gravity. We introduce
thin-shell wormholes that are categorized into a class of traversable wormhole
solutions. After that, we discuss stable thin-shell wormholes with
negative-tension branes in Reissner-Nordstr\"om-(anti) de Sitter spacetimes in
$d$ dimensional Einstein gravity. Imposing $Z_2$ symmetry, we construct and
classify traversable static thin-shell wormholes in spherical, planar and
hyperbolic symmetries. It is found that the spherical wormholes are stable
against spherically symmetric perturbations. It is also found that some classes
of wormholes in planar and hyperbolic symmetries with a negative cosmological
constant are stable against perturbations preserving symmetries. In most cases,
stable wormholes are found with the appropriate combination of an electric
charge and a negative cosmological constant. However, as special cases, there
are stable wormholes even with a vanishing cosmological constant in spherical
symmetry and with a vanishing electric charge in hyperbolic symmetry.
Subsequently, the existence and dynamical stability of traversable thin-shell
wormholes with electrically neutral negative-tension branes is discussed in
Einstein-Gauss-Bonnet theory of gravitation. We consider radial perturbations
against the shell for the solutions, which have the $Z_2$ symmetry. The effect
of the Gauss-Bonnet term on the stability depends on the spacetime symmetry.
