  $T$-adic exponential sums associated to a Laurent polynomial $f$ are
introduced. They interpolate all classical $p^m$-power order exponential sums
associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of
$T$-adic exponential sums is established. This bound enables us to determine,
for all $m$, the Newton polygons of $L$-functions of $p^m$-power order
exponential sums associated to an $f$ which is ordinary for $m=1$. Deeper
properties of $L$-functions of $T$-adic exponential sums are also studied.
Along the way, new open problems about the $T$-adic exponential sum itself are
discussed.
