Summary: "The last decades have seen an explosion of interest in the study of nonlinear dynamical systems. Many scientists realise the power of qualitative techniques developed during this period. The main idea is that the gross behaviour of all (at least main) solutions of the system is more important than the local behaviour of particular, analytically precise solutions. "Among these results is the well-known and surprising theorem of Sharkovski\u\i discussed here. In this paper, which is a summary of a talk given at Tribhuvan University in December 1993, we try to present an overview of the framework introduced these last 20 years to study the structure of the set of periodic orbits of discrete dynamical systems in dimensions 1 and 2."