  Ultrafunctions are a particular class of functions defined on a Non
Archimedean field E. They have been introduced and studied in some previous
works. In this paper we develop the notion of fine ultrafunctions which
improves the older definitions in many crucial points. Some applications are
given to show how ultrafunctions can be applied in studing Partial Differential
Equations. In particular, it is possible to prove the existence of
ultrafunction solutions to ill posed evolution poblems.
