We derive new sufficient conditions for uniform asymptotic stability of the zero solution of linear non-autonomous delay differential equations. The equations considered include scalar equations of the form
\[
x
′
(
t
)
=
−
c
(
t
)
x
(
t
)
+
∑
i
=
1
n
b
i
(
t
)
x
(
t
−
T
i
)
x’\left ( t \right ) = - c\left ( t \right )x\left ( t \right ) + \sum \limits _{i = 1}^n {{b_i}\left ( t \right )x\left ( {t - {T_i}} \right )}
\]
where
c
(
t
)
c\left ( t \right )
,
b
i
(
t
)
{b_i}\left ( t \right )
are continuous for
t
≥
0
t \ge 0
and
T
i
{T_i}
is a positive number
(
i
=
1
,
2
,
.
.
.
,
n
)
(i = 1, 2,...,n)
, and also systems of the form
\[
x
′
(
t
)
=
B
(
t
)
x
(
t
−
T
)
−
C
(
t
)
x
(
t
)
x’(t) = B(t)x(t - T) - C(t)x(t)
\]
where
B
(
t
)
B(t)
) and
C
(
t
)
C(t)
are
n
×
n
n \times n
matrices. The results are found by using the method of Lyapunov functionals.