Diophantine analysis of Markoff type surfaces

We review the rudiments of a diophantinetheory of affine Markoff cubics. These enjoy an action ofthe mapping class group on p-adic integral points thanks to their realization as the character varietyof the once punctured torus. This provides a powerful tool making them one of the few families of affine log K_3's whose integral points can be studied.We highlight recent advances and some applications.