Stability criteria in a nutshell

A continuous linear system is stable if and only if all its eigenvalues have negative real parts.

There are various other stability criteria, but they boil down to the statement above. For example, there is the Routh stability criterion and the Hurwitz stability criterion. There's also a continued fraction criterion.

But these criteria are just algorithms for determining whether the roots of the characteristic polynomial are all negative; they're not independent criteria. They were more important in the days of hand calculations; now it's easy enough to simply compute the roots of the characteristic polynomial numerically.

A discrete-time linear system is stable if and only if all its eigenvalues are inside the unit circle. The change of variables

w = (z - 1) / (z + 1)

maps the interior of the unit disk to the open left half plane, and so the roots of the characteristic polynomial in z are inside the unit circle if and only if the roots of the corresponding function of w are in the left half plane.

Related post: Cayley-Hamilton theorem