The subject of the paper is the autonomous delay monotone cyclic feedback system of the form $\dot x^0 (t) = f^0 (x^0(t), x^1(t))$, $\dot x^i (t) = f^i (x^{i - 1}(t)$, $x^i (t)$, $x^{i + 1} (t))$, $1 \le i \le N - 1$, $\dot x^N(t) = f^N (x^{N - 1}(t)$, $x^N (t)$, $x^0 (t - 1))$, where the functions $f^0$, $f^1, \dots, f^N$ satisfy some monotonicity conditions with respect to their first and last arguments. The authors show that under rather mild conditions on the nonlinearities $f^0$, $f^1, \dots,f^N$ the Poincaré-Bendixson theorem holds in force: either (a) the $\omega$-limit set $\omega (x)$ of a bounded solution $x$ is a single non-constant periodic orbit; or, else, (b) all $\alpha$- and $\omega$-limit points of any solution in $\omega (x)$ are equilibrium points of the system. The most part of the paper is devoted to the proof of this result. In Section 7 the authors investigate and provide very interesting results on the behavior of the periodic solutions with an emphasis on the winding number of the curve $t \to (x^i(t), x^{i + 1} (t))$ on the plane, while in the last section they examine the connection between oscillation of a periodic solution and its instability.