In the early 60's Sarkovskii discovered his famous theorem
on the coexistence of periodic orbits for interval maps.
Then, in the mid 70's, Li & Yorke rediscovered this result
and somewhat later the papers by Feigenbaum and Coullet & Tresser
on renormalisation and by Guckenheimer and Misiurewicz on
sensitive dependence and existence of invariant measures,
kicked off one of the most exciting areas within dynamical systems: iterations in dimension one.
The purpose of this paper is to survey some of the recent developments,
and pose some of the challenges and questions that keep this subject so intriguing.