  In this paper we study the linear series $|L-3p|$ of hyperplane sections with
a triple point $p$ on a surface $S$ embedded via a very ample line bundle $L$
for a \emph{general} point $p$. If this linear series does not have the
expected dimension we call $(S,L)$ \emph{triple-point defective}. We show that
on a triple-point defective surface through a general point every hyperplane
section has either a triple component or the surface is rationally ruled and
the hyperplane section contains twice a fibre of the ruling.
