It is well known that some solutions of non-linear partial differential equations (PDEs), like Einstein or Yang-Mills equations, exhibit linearization instability: some linearized solutions do not extend to families of near-by non-linear solutions. Often, linearized solution fail to extend when some non-linear functional, which we refer to as a linearization obstruction, is non-zero on it. In the case of Einstein and Yang-Mills equations, such linearization obstructions are precisely related to spacetime topology, charges of linearized conservation laws and rigid symmetries of the background solution. I will describe a significant generalization this classic result. It is applicable to both elliptic and hyperbolic equations, to variational and non-variational equations, to determined systems and gauge theories, and to ordinary as well as higher gauge theories.