Symmetries of a quantum field theory are implemented by topological operators. These are special extended operators whose correlation functions are insensitive to continuous deformations of their support. The classification of generalized symmetries thus reduces to understanding the spectrum of such topological operators across different codimensions. While a generic QFT may admit infinitely many topological operators, their topological nature imposes strong consistency conditions on their structure. In this talk, I will present a set of such constraints in 2+1 and 3+1 dimensions and highlight how they severely restrict the spectrum of topological operators in higher dimensions, in contrast to 1+1 dimensions. Using these constraints, I will argue that the action of non-invertible symmetries on local operators in higher dimensions is highly restricted. In particular, this action is either invertible or, when non-invertible, admits a description in terms of gauging a finite symmetry.