  The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$.
Our main result, is that on a surface of general type there are only finitely
many curves with negative self--intersection and sufficiently large canonical
degree. Our proof strongly relies on results by Miyaoka. We extend our result
both to surfaces not of general type and to non--negative curves, and give
applications, e.g. to finiteness of negative curves on a general blow--up of
$\mathbb P^ 2$ at $n\geq 10$ general points (a result related to \emph{Nagata's
Conjecture}). We finally discuss a conjecture by Vojta concerning the
asymptotic behaviour of the ratio between the canonical degree and the
geometric genus of a curve varying on a surface. The results in this paper go
in the direction of understanding the \emph{bounded negativity} problem.
