The encyclopedia article Hahn and Zermelo 1904, submitted in January 1904, is a continuation of Adolf Kneser's encyclopedia article 1904 which was actually finished already in 1900. It aims at closing the gap between 1900 and 1904 and reports about the progress in six sections, ranging from the foundations of Weierstrass's theory, necessary and sufficient conditions, isoperimetric problems, more general problems and examples and applications to existence questions.