  We construct automorphic representations for quasi-split groups $G$ over the
function field $F=k(t)$ one of whose local components is an epipelagic
representation in the sense of Reeder and Yu. We also construct the attached
Galois representations under the Langlands correspondence. These Galois
representations give new classes of conjecturally rigid, wildly ramified
$^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the
Kloosterman sheaves constructed earlier by Heinloth, Ng\^o and the author. We
study the monodromy of these local systems and compute all examples when $G$ is
a classical group.
