  A fundamental aspect of the three-body problem is its stability. Most
stability studies have focused on the co-planar three-body problem, deriving
analytic criteria for the dynamical stability of such pro/retrograde systems.
Numerical studies of inclined systems phenomenologically mapped their stability
regions, but neither complement it by theoretical framework, nor provided
satisfactory fit for their dependence on mutual inclinations. Here we present a
novel approach to study the stability of hierarchical three-body systems at
arbitrary inclinations, which accounts not only for the instantaneous stability
of such systems, but also for the secular stability and evolution through
Lidov-Kozai cycles and evection. We generalize the Hill-stability criteria to
arbitrarily inclined triple systems, explain the existence of quasi-stable
regimes and characterize the inclination dependence of their stability. We
complement the analytic treatment with an extensive numerical study, to test
our analytic results. We find excellent correspondence up to high inclinations
$(\sim120^{\circ}$), beyond which the agreement is marginal. At such high
inclinations the stability radius is larger, the ratio between the outer and
inner periods becomes comparable, and our secular averaging approach is no
longer strictly valid. We therefore combine our analytic results with
polynomial fits to the numerical results to obtain a generalized stability
formula for triple systems at arbitrary inclinations. Besides providing a
generalized secular-based physical explanation for the stability of non
co-planar systems, our results have direct implications for any triple systems,
and in particular binary planets and moon/satellite systems; we briefly discuss
the latter as a test case for our models.
