We consider a system of functional differential equations
x
′
(
t
)
=
F
(
t
,
x
(
⋅
)
)
x’\,(t)\, = \,\mathcal {F}\,(t,\,x( \cdot ))
, together with a Liapunov functional
V
(
t
,
x
(
⋅
)
)
\mathcal {V}\,(t,\,x( \cdot ))
with
V
′
⩽
0
\mathcal {V}’\, \leqslant \,0
. Most classical results require that
F
\mathcal {F}
be bounded for
x
(
⋅
)
x( \cdot )
bounded and that
F
\mathcal {F}
depend on
x
(
s
)
x(s)
only for
t
−
α
(
t
)
⩽
s
⩽
t
t\, - \,\alpha (t)\, \leqslant \,s\, \leqslant \,t
where
α
\alpha
is a bounded function in order to obtain stability properties. We show that if there is a function
H
(
t
,
x
)
H(t,\,x)
whose derivative along
x
′
(
t
)
=
F
(
t
,
x
(
⋅
)
)
x’\,(t)\, = \,\mathcal {F}\,(t,\,x( \cdot ))
is bounded above, then those requirements can be eliminated. The derivative of H may take both positive and negative values. This extends the classical theorem on uniform asymptotic stability, gives new results on asymptotic stability for unbounded delays and unbounded
F
\mathcal {F}
, and it improves the standard results on the location of limit sets for ordinary differential equations.