From the Galilean invariance of Newton’s mechanics, through gauge symmetries of Maxwell’s theory, Noether’s theorems, Lorentz’s invariance of special relativity, the principle of equivalence of general relativity, to gauge symmetries of the new unification theories, symmetries have played a central role in the research of nature. In this dissertation we will turn our attention to the interaction of spacetime symmetries and gauge or matter field symmetries in gravitational theories. After a mathematical introduction to symmetries, we will systematically present the sofar known results on scalar and linear electromagnetic fields symmetry inheritance, and show the latest progress made for the nonlinear electromagnetic fields. Everything shown will be put in context with no hair theorems for black holes and electromagnetic scalar potentials, where the inheritance is implicitly assumed. Finally, we will specify some solutions which do not inherit symmetries, to state the importance of making a classification of symmetry-breaking solutions.