This English translation of the author's original work has been thoroughly revised, expanded and updated.<P>The book covers logical systems known as <IT>type-free</IT> or <IT>self-referential</IT>. These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these theories provide a new outlook on classical topics, such as inductive definitions and predicative mathematics; (iii) they are particularly promising with regard to applications.<P>Research arising from paradoxes has moved progressively closer to the mainstream of mathematical logic and has become much more prominent in the last twenty years. A number of significant developments, techniques and results have been discovered.<P>Academics, students and researchers will find that the book contains a thorough overview of all relevant research in this field.