Recent results on functional equations in a single variable, perspectives and open problems

Summary.

In 1990 Cambridge University Press published the by now well-known monograph Iterative Functional Equations by Marek Kuczma, Bogdan Choczewski and Roman Ger. In the paper we will pay attention mainly on, we think, important papers published in the last decade or not discussed in this book. Of course, the selection we made reflects our personal preferences only.¶The topics of this paper are: 1. A view of the theory developed up to now; 2. Iterative roots; 3. Functional equations with superpositions of the unknown function; 4. Some linear and convolution equations; 5. Schilling's problem; 6. Daróczy's equation; 7. Simultaneous functional equations; 8. Schröder's equation; 9. Abel's equation; 10. Miscellaneous results and problems.