Necessary and sufficient conditions for oscillation of solutions of the equation $$ y'(t)+ \gamma f(t,y(t),y(\Delta_1(t,y(t))),\dots, y(\Delta_n(t,y(t))))= Q(t),\enskip t\geq t_0\in R,\enskip \gamma= \pm 1,\enskip n\geq 1 $$ are obtained in the case when $Q(t)\equiv 0$ on $[t_0,\infty)$ and sufficient conditions for oscillation and/or nonoscillation are obtained in the case when $Q(t)\not\equiv 0$ on $[t_0,\infty)$. The asymptotic behaviour of oscillatory and nonoscillatory solutions of this equation is studied, too.