  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.
