We discuss how a geometric approach to the symmetries of a critical system leads to novel predictions for the structure of correlators in generic d>2 bounded systems at criticality. Questions that can be tackled with this mindset are: How does the ordered state on the surface penetrate into a critical bulk? Does a two-point function depend fully on the position of both points or it actually depends on a reduced number of relevant variables? The answer will entail defining and solving a fractional version of the Yamabe problem making an anomalous curvature constant. The validity of this framework (which we dub Critical Geometry) is put to test in a wide and representative range of statistical models.