Abstract
Markoff triples are integer solutions to the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. We will review some of these and will then discuss recent joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions. We show in particular that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.