  The main goal of this project is to prove the equivalency of several
characterizations of completeness of Archimedean ordered fields; some of which
appear in most modern literature as theorems following from the Dedekind
completeness of the real numbers, while a couple are not as well known and have
to do with other areas of mathematics, such as nonstandard analysis.
  Continuing, we study the completeness of non-Archimedean fields, and provide
several examples of such fields with varying degrees of properties, using
nonstandard analysis to produce some relatively "nice" (in particular, they are
Cantor complete) final examples.
  As a small detour, we present a short construction of the real numbers using
methods from nonstandard analysis.
