An aspect of mathematics that most excites and interests its proponents is the way in which seemingly quite disparate areas interact. These often deep and subtle connections can give rise to significant advances in the subject. The work of A. A. Markoff, published in 1879 and 1890, was ostensibly on the normalized infima of values of indefinite real quadratic forms on the nonzero points of the integer lattice. The set of these infima is now known as the Markoff spectrum. The problem that Markoff studied was recognized as being in the area of diophantine approximation and his methods reflected this fact. He used continued fractions to describe completely those quadratic forms whose normalized infima lie above 1/3. The choice of 1/3 was not an accident. The corresponding forms have a remarkable structure not possessed by forms with normalized infima lower down the Markoff spectrum.