Let X_n be the set of conjugacy classes of n-tuples of 2x2 matrices whose product is the identity matrix. There is a natural braid group action on X_n, whose study goes back to work of Markoff in the late 19th century. The most basic question one can ask about this action, which dates to work of Painlevé, Fuchs, Schlesinger, and Garnier in the beginning of the 20th century, is: what are the finite orbits of this action? I'll explain the history of this question, as well as some recent work, joint with Lam and Landesman, in which we give a complete classification of such finite orbits, by algebro-geometric methods, when at least one of the matrices in question has infinite order. Time permitting, I'll discuss other variants of this question, whose answer relies on non-abelian Hodge theory and the Langlands program, and resolves conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang.