Many spaces of multivariate piecewise functions (or splines) with prescribed global smoothness exhibit additional, unprescribed, or intrinsic supersmoothness at certain faces where the pieces join together. Intrinsic supersmoothness is a consequence of the geometry of the underlying partition. In this survey, we provide insights into this phenomenon, review and generalize known results, and demonstrate how intrinsic supersmoothness affects other properties of spline spaces.