This book presents an exceptional exposition of Gödel's incompleteness theorems for non-specialists. The author explains clearly and thoroughly, without too many technical details, what the theorems really say and also, as the title of the book suggests, what they do not. The book corrects frequent misunderstandings and misapplications to various areas like philosophy, physics and theology. It also presents some interesting historical comments on the discovery of the theorems. Finally, it briefly describes examples of some more recent results revealed by Gödel's incompleteness theorems. Among others, the connections between incompleteness and Kolmogorov complexity, a notion from the area of algorithmic randomness, and the Paris-Harrington theorem are briefly discussed. To summarize, it is a valuable addition to the literature.