We shall discuss some famous problems which occupied the attention of mathematicians for thousands of years and which have been shown to be unsolvable in recent decades and, in addition, some very old classical problems which, up till now, have scarcely been mentioned in books on the subject. We have endeavoured to vary well-known methods of proof for classical problems and to generalise their formulation.Thus, the object of this book is not the systematic representation of a well-defined section of geometry. But rather our aim is of a methodological nature: we want to develop the kind of arguments which are suitable for the solution of the geometrical problems considered and to discuss the questions implied by the existence of unsolvable problems.In the experience of mathematical institutes and authors of mathematical publications attempts are still made to solve problems which are known to be unsolvable. And so we fear that there will be some readers who, after reading this book, will believe that, despite everything, they have found a solution to the problem of trisecting an angle. We should like to advise these readers as follows: Read through the proof showing that such a solution is impossible three times and then try to find the mistake in your argument. Better still, give up such attempts since they really cannot lead anywhere. There are plenty of open questions—many of which are stated in this book—to which mathematical intuition may be applied with real prospect of success. However, it should be noted that it is not particularly easy to solve the open questions stated. It is sometimes much simpler to discover new results in a new branch of mathematics rather than to solve one of the problems left open in elementary geometry. However, according to ErhardSchmidt, it is better to solve old problems with new methods than to solve new problems with old. Therefore we feel justified in encouraging work on the “ elementary ” problems stated in this book.For the reader’s convenience there is, at the end of the Table of Contents, a diagram showing the interrelationships of the chapters.