  The purpose of the present paper is to investigate $G$-opers on pointed
Riemann surfaces (for a simple algebraic group $G$ of adjoint type) and their
monodromy maps. In the first part, we review some general facts on $G$-opers,
or more generally, principal $G$-bundles with holomorphic connection having
simple poles along marked points, including the correspondence with
$G$-representations of the fundamental group. One of the main results, proved
in the second part, asserts that the space of certain $G$-opers with real
monodromy forms a discrete set. This fact generalizes the discreteness theorem
for real projective structures, already proved by G. Faltings. As an
application, we establish the Eichler-Shimura isomorphism for each
$\mathrm{PSL}_2$-oper with real monodromy. The resulting decomposition of the
(parabolic) de Rham cohomology group of its symmetric product defines a
polarized real Hodge structure.
