Following the congresses in Paris 1992, Budapest 1996 and Barcelona 2000, the Fourth European Congress of Mathematics took place in Stockholm in July 2004. There were seven plenary lectures, 33 invited lectures, twelve European network lectures, six science lectures and 322 poster presentations covering all areas of mathematics and its various applications. The proceedings includes 32 contributions by invited speakers, eleven presentations by the EU Research Training Networks in Mathematics and Information Sciences and programmes from the European Science Foundation in Physical and Engineering Science and all seven papers by the plenary speakers. Furthermore, five contributions by EMS prize winners are included in the proceedings. There is also a foreword by the president of the EMS, the opening speech and scientific report by the president of the 4ECM Organization Committee. This book represents a nice overview of many fields of contemporary mathematics.