A glance of the table of contents will reveal that this research book treats topics in analysis at the "Advanced fractional di erential equations" level.The aim has been to provide a development of the subject which is honest,rigorous,up to date and at the same time, not too pedantic. The book provides a transition from elementary fractional di erential equations to advanced courses in di eren- tial equations and it introduces the reader to some of the abstract thinking that pervades modern analysis.