  In this paper we study the linear series |L-3p| of hyperplane sections with a
triple point p of a surface S embedded via a very ample line bundle L for a
general point p. If this linear series does not have the expected dimension we
call (S,L) triple-point defective. We show that on a triple-point defective
regular 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.
