This paper deals with a new kind of generalized functions, called "ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedeanfield, 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 applications of this kind will be presented in the second part of this paper.