Branch curves and adjoint curves

In 1929, Zariski has found that the branch curve of a smooth cubic surface in P^3
(over an algebraically closed field of char=0) is a sextic plane curve with 6 cusps, all of them lying on a conic.
A year later, Segre generalized this, proving a similar theorem on smooth surfaces of any degree in P^3. Explicitly, he proved that there are two curves of unexpectedly low degree, passing through the
nodes and the cusps of the branch curve of this surface. We will discuss these theorems, and also their generalizations to any surface in P^N. We also discuss the notion of Zariski pairs, induced from the reduciblility of some of the varieties of nodal-cuspidal plane curves.