The abstract of the talk (see below) may be downloaded in one of the following formats:
The celebrated Poincar\'e-Bendixson Theorem assures that given a $C^1$-flow on $\mathbb{S}^2$, $\Psi:\mathbb{R}\times\mathbb{S}^2\rightarrow\mathbb{S}^2$, and a point $x$ whose $\omega$-limit set does not contains any singular point, then $\omega_\Phi(x)$ is a closed orbit. The aim of this talk is to show in which phase spaces this Theorem also works, in particular we will see that it is valid for the Klein Bottle and the Projective Plane with the same formulation that in the case of $\mathbb{S}^2$. In all the compact and connected surfaces, except for the Torus, we will see that the Theorem works for flows of $C^2$-class. Finally we will state the recent results in the Poincar\'e-Bendixson Theory, namely we will state topological characterizations of the $\omega$-limit sets in some phase spaces: Klein Bottle, Projective Plane and $n$-dimensional sphere.