The centenary of A.N.Kolmogorov, one of the greatest scientists of the 20th century, falls this year, 2003. He was born in Russia on the 25th of April 1903.1This is typically the occasion for apologetic portraits or hagiographic surveys about such an intense human and scientific biography. Various meetings and publications will be devoted to celebrate the work and the character of the great mathematician. So one could wonder why pubblishing a book which simply aims at popularizing his major achievements in fields out of pure mathematics? We are deeply convinced that Kolmogorov’s contributions are the cornerstone over which many modern research fields, from physics to computer science and biology, are based and still keep growing. His ideas have been transmitted also by his pupils to generations of scientists. The aim of this book is to extend such knowledge to a wider audience, including cultivated readers, students in scientific disciplines and active researchers.