  Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. Let $\nu
\in \text{Aut}\, \mathfrak{g}$ be a diagram automorphism whose order divides $T
\in \mathbb{Z}_{\geq 1}$. We define cyclotomic $\mathfrak{g}$-opers over the
Riemann sphere $\mathbb{P}^1$ as gauge equivalence classes of
$\mathfrak{g}$-valued connections of a certain form, equivariant under actions
of the cyclic group $\mathbb{Z}/ T\mathbb{Z}$ on $\mathfrak{g}$ and
$\mathbb{P}^1$. It reduces to the usual notion of $\mathfrak{g}$-opers when $T
= 1$.
  We also extend the notion of Miura $\mathfrak{g}$-opers to the cyclotomic
setting. To any cyclotomic Miura $\mathfrak{g}$-oper $\nabla$ we associate a
corresponding cyclotomic $\mathfrak{g}$-oper. Let $\nabla$ have residue at the
origin given by a $\nu$-invariant rational dominant coweight
$\check{\lambda}_0$ and be monodromy-free on a cover of $\mathbb{P}^1$. We
prove that the subset of all cyclotomic Miura $\mathfrak{g}$-opers associated
with the same cyclotomic $\mathfrak{g}$-oper as $\nabla$ is isomorphic to the
$\vartheta$-invariant subset of the full flag variety of the adjoint group $G$
of $\mathfrak{g}$, where the automorphism $\vartheta$ depends on $\nu$, $T$ and
$\check{\lambda}_0$. The big cell of the latter is isomorphic to $N^\vartheta$,
the $\vartheta$-invariant subgroup of the unipotent subgroup $N \subset G$,
which we identify with those cyclotomic Miura $\mathfrak{g}$-opers whose
residue at the origin is the same as that of $\nabla$. In particular, the
cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is
interpreted as taking $\nabla$ to other cyclotomic Miura $\mathfrak{g}$-opers
corresponding to elements of $N^\vartheta$ associated with simple root
generators.
  We motivate the introduction of cyclotomic $\mathfrak{g}$-opers by
formulating two conjectures which relate them to the cyclotomic Gaudin model of
[arXiv:1409.6937].
