In classical curve theory, the geometry of a curve in three dimen- sions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of nding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices. This paper shows constructively how to solve the natural equations explic- itly for an innite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant he- lices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices. The present paper gives for the rst time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix.