In this paper we are concerned with the asymptotic stability of the delay differential equation
\[
x
′
(
t
)
=
A
0
x
(
t
)
+
∑
k
=
1
n
A
k
x
(
t
τ
k
)
,
x^{\prime }(t)=A_0x(t)+\sum _{k=1}^nA_kx(t_{\tau _k}),
\]
where
A
0
,
A
k
∈
C
d
×
d
A_0,A_k\in C^{d\times d}
are constant complex matrices, and
x
(
t
τ
k
)
=
(
x
1
(
t
−
τ
k
1
)
,
x
2
(
t
−
τ
k
2
)
,
…
,
x
d
(
t
−
τ
k
d
)
)
T
,
τ
k
l
>
0
x(t_{\tau _k})= (x_1(t-\tau _{k1}),x_2(t-\tau _{k2}),\dots ,x_d(t-\tau _{kd}))^T,\tau _{kl}>0
stand for
n
×
d
n\times d
constant delays
(
k
=
1
,
…
,
n
,
l
=
1
,
…
,
d
)
(k=1,\dots ,n,l=1,\dots ,d)
. We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation
\[
x
′
(
t
)
=
L
x
(
t
)
+
∑
i
=
1
m
M
i
x
(
t
−
τ
i
)
+
∑
j
=
1
n
N
j
x
′
(
t
−
τ
j
′
)
,
x^{\prime }(t)=Lx(t)+\sum _{i=1}^mM_ix(t-\tau _i)+\sum _{j=1}^nN_jx^{\prime }(t-\tau _j^{\prime }),
\]
where
L
,
M
i
,
L,M_i,
and
N
j
∈
C
d
×
d
N_j\in C^{d\times d}
are constant complex matrices and
τ
i
,
τ
j
′
>
0
\tau _i,\tau _j^{\prime }>0
stands for constant delays
(
i
=
1
,
…
,
m
(i=1,\dots ,m
,
j
=
1
,
…
,
n
)
j=1,\dots ,n)
. Numerical examples on various circumstances are shown to check our results which are more general than those already reported.