  This expository paper is based on the author's series of lectures delivered
at the January 1999 Mini-course in Number Theory, held at Sogang University
(Seoul). The aim is to give an elementary and self-contained introduction to
the theory of Newton polygons (namely, the $p$-adic Riemann hypothesis) for
L-functions of exponential sums over a finite field.
  The main idea of our approach is to establish a suitable local to global
principal which would allowus to reduce the harder ``global'' case to various
easier well understood "local" cases for which Stickelberger's theorem applies.
For this purpose, a systematic decomposition method was introduced in our
earlier paper and several decomposition theorems were proved, including the
facial decomposition theorem, the star decomposition theorem and the hyperplane
decomposition theorem. These theorems easily recover previously known results
on Newton polygons of exponential sums. They are also enough for a number of
further applications such as Mazur's conjecture for a generic hypersurface and
a weaker form of the more general Adolphson-Sperber conjecture for a generic
exponential sum. Using our earlier method, we have recently found a more
flexible collapsing decomposition theorem. This theorem gives a fairly
satisfactory answer to the full form of the Adolphson-Sperber conjecture. In
particular, as an application, we now know that the full form of the
Adolphson-Sperber conjecture is true in every low dimension $n\leq 3$ but false
in every high dimension $n\geq 4$. These results are explained in these notes.
