  We define and study a relative perverse $t$-structure associated with any
finitely presented morphism of schemes $f: X\to S$, with relative perversity
equivalent to perversity of the restrictions to all geometric fibres of $f$.
The existence of this $t$-structure is closely related to perverse
$t$-exactness properties of nearby cycles. This $t$-structure preserves
universally locally acyclic sheaves, and one gets a resulting abelian category
$\mathrm{Perv}^{\mathrm{ULA}}(X/S)$ with many of the same properties familiar
in the absolute setting (e.g., noetherian, artinian, compatible with Verdier
duality). For $S$ connected and geometrically unibranch with generic point
$\eta$, the functor $\mathrm{Perv}^{\mathrm{ULA}}(X/S)\to
\mathrm{Perv}(X_\eta)$ is exact and fully faithful, and its essential image is
stable under passage to subquotients. This yields a notion of "good reduction"
for perverse sheaves.
