When is an object "random"? Kolmogorov (and others) argue that one could measure randomness by the shortest description, i.e. computer program, that generates it. This simple idea leads to a beautiful mathematical theory and a powerful tool as one can show that random objects have several interesting properties. Li and Vitanyi have written this wonderful monograph on the area covering the depth of theory and applications not seen anywhere else. They give a clear and complete descriptions of many of the important concepts in the book. I have used this book twice in teaching graduate courses on the topic. This book is a must have for anyone interested in a serious mathematical treatment of Kolmogorov complexity.
Aswell as an excellent textbook, this is a reference book for researchers. The third edition has recent updates of the progress that has been made in the field of Kolmogorov complexity.
The best thing about Li and Vitanyi's book is that it is very interesting. For example, it contains quotes about randomness from famous researchers like Laplace which deepens the reader's understanding of Kolmogorov complexity and makes the topic of randomess even more fascinating.
The book explains subjects very well. Many readers will gain a much deeper understanding of the areas of physics covered in this book than they would from a physics textbook. The book has interesting backgrounds to all the subjects which are covered in the book.