Continued fractions and the Markoff tree

We give here a full account of Markoffʹs celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohnʹs method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.