This book provides a negative solution to Zeuthen's problem, which was proposed as a prize problem in 1901 by the Royal Danish Academy of Arts and Sciences. The problem was to decide whether every irreducible family of smooth space curves admits limit curves which are stick figures, composed of lines meeting only two at a time. To solve the problem, the author makes a detailed study of curves on cubic surfaces in ${mathbb P}^3$ and their possible degenerations as the cubic surface specializes to a quadric plus a plane or the union of three planes.