  Let $C$ be an irreducible projective plane curve in the complex projective
space ${\mathbb{P}}^2$. The classification of such curves, up to the action of
the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very
difficult open problem with many interesting connections. The main goal is to
determine, for a given $d$, whether there exists a projective plane curve of
degree $d$ having a fixed number of singularities of given topological type. In
this note we are mainly interested in the case when $C$ is a rational curve.
The aim of this article is to present some of the old conjectures and related
problems, and to complete them with some results and new conjectures from the
recent work of the authors.
