Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday

The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer, Boston - Dordrecht - London, 2004) but a fresh look on things is necessary because many new contributions have been accumulated in the meantime. With no claim that our investigation is exhaustive we consider various directions of interest, try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems of acute interest.