  The classical $n$-variable Kloosterman sums over finite fields are well
understood by Deligne's theorem from complex point of view and by Sperber's
theorem from $p$-adic point of view. In this paper, we study the complex and
$p$-adic estimates of inverted $n$-variable Kloosterman sums, addressing a
question of N. Katz (1995). We shall give two complex estimates. The first one
is elementary based on Gauss sums. The second estimate is deeper, depending on
the cohomological results of Adolphson-Sperber, Denef-Loeser and Fu for twisted
toric exponential sums. This deeper result assumes that the characteristic $p$
does not divide $n+1$. Combining with Dwork's $p$-adic theory, we also
determine the exact $p$-adic valuations for zeros and poles of the L-function
associated to inverted $n$-variable Kloosterman sums in the case $p \equiv 1
\mod (n+1)$. As we shall see, the inverted $n$-variable Kloosterman sum is more
complicated than the classical $n$-variable Kloosterman sum in all aspects in
the sense that our understanding is less complete, partly because the Hodge
numbers are now mostly $2$ instead of $1$.
