Markov numbers in algebra, geometry and arithmetic

Markov's equation is the Diophantine equation
$$ x^2+y^2+z^2=3xyz. $$
It was first considered by A. A. Markov in connection with the theory of integer quadratic forms and diophantine approximations. Later it turned out that solutions to the Markov equation suddenly appear in various, far from each other areas of mathematics. The talk will be devoted to an overview of solved and open problems related to the Markov equation and its generalizations.