The height of mountain ranges reflects the balance between tectonic rock uplift, crustal strength and surface denudation. Tectonic deformation and surface denudation are interdependent, however, and feedback mechanisms-in particular, the potential link to climate-are subjects of intense debate. Spatial variations in fluvial denudation rate caused by precipitation gradients are known to provide first-order controls on mountain range width, crustal deformation rates and rock uplift. Moreover, limits to crustal strength are thought to constrain the maximum elevation of large continental plateaus, such as those in Tibet and the central Andes. There are indications that the general height of mountain ranges is also directly influenced by the extent of glaciation through an efficient denudation mechanism known as the glacial buzzsaw. Here we use a global analysis of topography and show that variations in maximum mountain height correlate closely with climate-controlled gradients in snowline altitude for many high mountain ranges across orogenic ages and tectonic styles. With the aid of a numerical model, we further demonstrate how a combination of erosional destruction of topography above the snowline by glacier-sliding and commensurate isostatic landscape uplift caused by erosional unloading can explain observations of maximum mountain height by driving elevations towards an altitude window just below the snowline. The model thereby self-consistently produces the hypsometric signature of the glacial buzzsaw, and suggests that differences in the height of mountain ranges mainly reflect variations in local climate rather than tectonic forces.