God not only plays dice in quantum mechanics, but even with the whole numbers! The discovery of randomness in arithmetic is presented in my book Algorithmic Information Theory published by Cambridge University Press. There I show that to decide if an algebraic equation in integers has finitely or infinitely many solutions is in some cases absolutely intractable. I exhibit an infinite series of such arithmetical assertions that are random arithmetical facts, and for which it is essentially the case that the only way to prove them is to assume them as axioms. This extreme form of Gdel incompleteness theorem shows that some arithmetical truths are totally impervious to reasoning. The papers leading to this result were published over a period of more than twenty years in widely scattered journals, but because of their unity of purpose they fall together naturally into the present book, intended as a companion volume to my Cambridge University Press monograph. I hope that it will serve as a stimulus for work on complexity, randomness and unpredictability, in physics and biology as well as in metamathematics. For the second edition, I have added the article Randomness in arithmetic (Part I), a collection of abstracts (Part VII), and, as an Epilogue, two essays which have not been published elsewhere that assess the impact of algorithmic information theory on mathematics and biology, respectively. I should also like to point out that it is straightforward to apply to LISP the techniques used in Part VI to study bounded-transfer Turing machines. A few footnotes have been added to Part VI, but the subject richly deserves book length treatment, and I intend to write a book about LISP in the near future.