We define and study a relative perverse
t
t
-structure associated with any finitely presented morphism of schemes
f
:
X
→
S
f: X\to S
, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of
f
f
. The existence of this
t
t
-structure is closely related to perverse
t
t
-exactness properties of nearby cycles. This
t
t
-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category
P
e
r
v
U
L
A
(
X
/
S
)
\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
S
connected and geometrically unibranch with generic point
η
\eta
, the functor
P
e
r
v
U
L
A
(
X
/
S
)
→
P
e
r
v
(
X
η
)
\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.