One particularly important class of integrable systems are the Painleve/Garnier/Schlesinger-type systems that arise as "monodromy-preserving" flows in families of linear differential equations. It turns out that one can translate these flows into essentially geometric terms, in which the equations are sheaves on a suitable surface, and the (discrete) flows are twisting by a line bundle, with one caveat: the surface involved is noncommutative. I'll explain how this works, and sketch a number of consequences, e.g., how derived equivalences between noncommutative elliptic surfaces give rise to new isomonodromic descriptions for the Painleve equations.