  This paper deals with a new kind of generalized functions, called
"ultrafunctions" which have been introduced recently and developed in some
previous works. Their peculiarity is that they are based on a Non-Archimedean
field namely on a field which contains infinite and infinitesimal numbers.
Ultrafunctions have been introduced to provide generalized solutions to
equations which do not have any solutions not even among the distributions.
Some of these applications will be presented in the second part of this paper.
