The idea of writing some detailed notes on the Dirac operator, and possibly a monograph, came to my mind after I was invited to give a series of graduate lectures at SISSA by Mauro Carfora, in April 1994. Indeed, I had been a graduate student of Cambridge University at St. John's College from 1987 to 1991, sharing my time between the Department of Applied Mathematics and Theoretical Physics and St. John's College, working under the supervision of a distinguished scientist who had, himself, taken great inspiration from Dirac's work. Further to this, the main original results of my Ph.D. thesis were obtained within the framework of local or non-local boundary conditions for the Dirac operator, with application to heat-kernel asymptotics and one-loop quantum cosmology. The problem of boundary conditions in the formulation of physical laws, the interplay of geometric and analytic concepts, and the application of spinorial techniques have always been at the heart of my research since those early days.

Hence I felt it was time to revise substantially my original SISSA lecture notes (early versions appeared as reports GR-QC 9507046 and HEP-TH 9704016) and turn them into a book in the form of a lecture notes monograph for Ph.D. students and research workers. The expert reader will recognize the deep influence of my main sources. Indeed, it is the author's personal opinion that the best way to understand current developments in mathematics and physics is to follow the presentation given, in the original papers, by the leading authors in the various fields.