  We offer an axiomatic definition of a differential algebra of generalized
functions over an algebraically closed non-Archimedean field. This algebra is
of {\em Colombeau type} in the sense that it contains a copy of the space of
Schwartz distributions. We study the uniqueness of the objects we define and
the consistency of our axioms. The article is aimed at mathematicians and
physicists who are interested in the non-linear theory of generalized
functions, but who are not necessarily familiar with the original Colombeau
theory. We assume, however, some basic familiarity with the Schwartz theory of
distributions.
