From Plato, who denied entry into the Academy to all those who knew nothing of geometry, to Galileo Galilei´s statement that "The Universe ... is written in mathematical language ... ", mathematicians have developed an incomparable body of doctrine. This mathematical knowledge is constantly being transferred to other sciences, as well as contributing to the development of new technologies. To paraphrase Leonardo da Vinci, 'a science cannot be considered as such unless it is impregnated with mathematics'.

That is why mathematics has become an extraordinary instrument of development for peoples, and has been recognized as such the world over. But mathematics is not only useful; they are also the foremost expression of the human spirit, and the greatest intellectual challenge for humanity. Mathematicians are addressing the biggest problems confronting their discipline with the determination of overcoming them, or as the German mathematician Gustav Jacobi expressed it, 'in honour of the human spirit'.

Mathematical research at an international level is a recent phenomenon in Spain. In the last 20 years, Spanish mathematical production as recorded in the Web of Knowledge journals has jumped from a mere token presence to a figure of 5%, making it the third-ranking science at a national level. Today, Spain is a very 'mathematized' country, while the Spanish mathematical community has attained a highly articulated structure. As a result of this spectacular development, at the 2002 General Assembly in Shangai, the International Union of Mathematics decided to make Spain the host of the International Congress of Mathematicians for the first time in its history. This outstanding event will take place from August 22nd to 30th in Madrid, and will be the greatest ever to have been held in the history of Spanish science.

The International Congress of Mathematicians is more than just another congress; mathematicians from all over the world will gather in Madrid to honour the finest achievements of their colleagues with the award of the Fields Medals, the Nevalinna Medal and the Gauss Medal. Furthermore, over the 10 days of the congress, the latest advances will be presented, the burning issues of the day in mathematical research, applications and education will be discussed, and prospects for mathematics in the 21st century will be outlined.

The Ramón Areces Foundation and the Royal Academy of Sciences, together with the Executive Committee of the ICM2006 Madrid, wish to contribute to this event with the organization of a symposium to which leading international figures in mathematics and scientists from other disciplines are invited. The aim is to discuss what kind of mathematics we need to tackle successfully the challenges arising in the current century.

This overview will range from classical problems such as Poincaré´s conjecture, probably one of the central issues to be dealt with at the ICM2006, to the relations between mathematics and biology, not forgetting relations with computing, engineering, industry and finance. And lastly, but by no means least, the ongoing, two-way connection with the physical sciences, which has done so much to shape the modern world and our vision of the universe.