  Deligne constructed a remarkable local system on $\bP^1-\{0,\infty\}$
attached to a family of Kloosterman sums. Katz calculated its monodromy and
asked whether there are Kloosterman sheaves for general reductive groups and
which automorphic forms should be attached to these local systems under the
Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we find
an explicit family of such automorphic forms and even a simple family of
automorphic sheaves in the framework of the geometric Langlands program. We use
these automorphic sheaves to construct l-adic Kloosterman sheaves for any
reductive group in a uniform way, and describe the local and global monodromy
of these Kloosterman sheaves. In particular, they give motivic Galois
representations with exceptional monodromy groups G_2,F_4,E_7 and E_8. This
also gives an example of the geometric Langlands correspondence with wild
ramifications for any reductive group.
