  We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary
finitely generated group. Strong quasiconvexity generalizes quasiconvexity in
hyperbolic groups and is preserved under quasi-isometry. We show that strongly
quasiconvex subgroups are also more reflexive of the ambient groups geometry
than the stable subgroups defined by Durham-Taylor, while still having many
analogous properties to those of quasiconvex subgroups of hyperbolic groups. We
characterize strongly quasiconvex subgroups in terms of the lower relative
divergence of ambient groups with respect to them.
  We also study strong quasiconvexity and stability in relatively hyperbolic
groups, right-angled Coxeter groups, and right-angled Artin groups. We give
complete descriptions of strong quasiconvexity and stability in relatively
hyperbolic groups and we characterize strongly quasiconvex special subgroups
and stable special subgroups of two dimensional right-angled Coxeter groups. In
the case of right-angled Artin groups, we prove that two notions of strong
quasiconvexity and stability are equivalent when the right-angled Artin group
is one-ended and the subgroups have infinite index. We also characterize
non-trivial strongly quasiconvex subgroups of infinite index (i.e. non-trivial
stable subgroups) in right-angled Artin groups by quadratic lower relative
divergence, expanding the work of Koberda-Mangahas-Taylor on characterizing
purely loxodromic subgroups of right-angled Artin groups.
