  Starting from a finite simple graph $G$, for each eigenvalue $\theta$ of its
adjacency matrix one can construct a convex polytope $P_G(\theta)$, the so
called $\theta$-eigenpolytop of $G$. For some polytopes this technique can be
used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall
call them spectral) are still badly understood. We give an overview of the
literature for eigenpolytopes and spectral polytopes.
  We introduce a geometric condition by which to prove that a given polytope is
spectral (more exactly, $\theta_2$-spectral). We apply this criterion to the
edge-transitive polytopes. We show that every edge-transitive polytope is
$\theta_2$-spectral, is uniquely determined by this graph, and realizes all its
symmetries. We give a complete classification of distance-transitive polytopes.
