Necessary, sufficient, and necessary and sufficient conditions are obtained for all solutions of the nonlinear differential equation
\[
d
y
d
t
+
∑
j
=
1
n
q
j
f
(
y
(
t
−
τ
j
)
)
=
0
,
t
≥
0
,
(
∗
)
\frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}f\left ( {y\left ( {t - {\tau _j}} \right )} \right ) = 0,\qquad t \ge 0,} \qquad \left ( * \right )
\]
to be oscillatory. These conditions are expressed in terms of the characteristic equation of the corresponding linear “variational” equation
\[
d
y
d
t
+
∑
j
=
1
n
q
j
y
(
t
−
τ
j
)
=
0
,
t
≥
0.
(
∗
∗
)
\frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}y\left ( {t - {\tau _j}} \right ) = 0, \qquad t \ge 0. \qquad \left ( { * * } \right )}
\]
Our results show that for a certain class of nonlinear functions
f
,
(
∗
)
f,\left ( * \right )
oscillates if and only if
(
∗
∗
)
\left ( { * * } \right )
oscillates. As an application of our results, we obtain simple sufficient and necessary and sufficient conditions for the oscillation of several nonlinear delay differential equations which appear in applications.