Arithmetic and Dynamics on Varieties of Markoff Type

The Markoff equation x2+y2+z2=3xyz, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts.  After reviewing some of these, we will discuss joint work with Bourgain and Sarnak establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.