This book arose from many lectures the authors delivered independently at different locations to students of different levels.

‘Theory’ is a scientific name for ‘story’. So, if the reader somehow feels uncomfortable about following a theory of continued fractions, he or she might be more content to read the story of neverending fractions.

The queen of mathematics – number theory – remains one of the most accessible parts of significant mathematical knowledge. Continued fractions form a classical area within number theory, and there are many textbooks and monographs devoted to them. Despite their classical nature, continued fractions remain a neverending research field, many of whose results are elementary enough to be explained to a wide audience of graduates, postgraduates and researchers, as well as teachers and even amateurs in mathematics. These are the people to whom this book is addressed.

After a standard introduction to continued fractions in the first three chapters, including generalisations such as continued fractions in function fields and irregular continued fractions, there are six ‘topics’ chapters. In these we give various amazing applications of the theory (irrationality proofs, generating series, combinatorics on words, Somos sequences, Diophantine equations and many other applications) to seemingly unrelated problems in number theory. The main feature that we would like to make apparent through this book is the naturalness of continued fractions and of their expected appearance in mathematics. The book is a combination of formal and informal styles. The aforementioned applications of continued fractions are, for the most part, not to be found in earlier books but only in scattered scientific articles and lectures.