  We present a method which provides a unified framework for most stability
theorems that have been proved in graph and hypergraph theory. Our main result
reduces stability for a large class of hypergraph problems to the simpler
question of checking that a hypergraph $\mathcal H$ with large minimum degree
that omits the forbidden structures is vertex-extendable. This means that if
$v$ is a vertex of $\mathcal H$ and ${\mathcal H} -v$ is a subgraph of the
extremal configuration(s), then $\mathcal H$ is also a subgraph of the extremal
configuration(s). In many cases vertex-extendability is quite easy to verify.
  We illustrate our approach by giving new short proofs of hypergraph stability
results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene
Watts-Norin-Yepremyan and others. Since our method always yields minimum degree
stability, which is the strongest form of stability, in some of these cases our
stability results are stronger than what was known earlier. Along the way, we
clarify the different notions of stability that have been previously studied.
