A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain symmetries), spectral graph theory (satisfying certain self-stress conditions) and rigidity theory (admitting deformations that do not alter the symmetry properties). We explore the connections between these perspectives, with a focus on realizations of highly symmetric graphs (arc-transitive/distance-transitive) and the question of how much symmetry is necessary to ensure that a realization is balanced, spectral, rigid etc. We include many examples to give a broad overview of the possibilities and restrictions of symmetric and spectral graph realizations.