Markoff Equation

Studying approximations of real numbers by rational numbers, A.A. Markov introduced a new Diophantine equation in 1879:
$$x^2 + y^2 + z^2 = 3xyz,$$
which later became known as the Markoff equation. Set of its natural solutions, the "Markoff triples", has a natural tree-graph structure. In recent years, influenced by the work of Bourgain, Gamburd, and Sarnak, the Markoff equation has come to be studied over the field of residues modulo prime $p$. Last year, Chen published the completion of a very complex proof of the main conjecture, which states that for sufficiently large primes $p$, all solutions of the Markov equation over the field of residues modulo $p$ are obtained from its integer solutions by reduction modulo $p$. The proof of the conjecture is based on several papers using very different methods.
I plan to discuss these facts, including Markov's classical results, as well as completely new generalizations to the $n$-dimensional case.