  A joint generalization of real smooth as well of complex manifolds are the
Cauchy-Riemann manifolds. The main objective of the paper is to inroduce a
class of symmetric CR manifolds containing both classes of Riemannian and
Hermitian symmetric spaces. It turns out that the classical requirement of
isolated fixed points for the symmetries is no longer adequate, because it
would imply the Levi-flatness.
  Among all symmetric CR-manifolds we distinguish a large subclass consisting
of all Shilov boundaries of bounded symmetric domains. For this class we
calculate the polynomial and the rational convex hulls Both hulls are
canonically stratified into real-analytic CR-submanifolds such that the
(unique) stratum of the highest dimension is complex for the polynomial and
Levi-flat for the rational convex hull. It is also proved that the CR-functions
extend continuously to the rational convex hull such that the extension is CR
on every stratum.
