In this third edition the book has been expanded in three directions: 1. Problems have been added at the end of each chapter (40% of which are solved in the last section of the book) together with suggestions for further reading. Furthermore, the number of appendices (marked with a grey stripe) has been substantially enlarged in order to make the book more self-su?cient. These additions, together with many clari?cations in the text, render the book more suitable as a companion in a course on Green's functions and their applications. 2. Theimpressivedevelopmentsofthe1980sand1990sinmesoscopicphysics, and in particular in transport properties, found their way - to a c- tain extent - in the new Chaps.8 and 9 (which also contain some of the material of the old Chap.7). This is a natural expansion, since Green's functions have played an important role as a theoretical tool in this new ?eld of physics, a role that continues in nanoregime research (see, e.g., recent publications dealing with carbon nanotubes). Thus, the powerful and unifying formalism of Green's functions ?nds applications not only in standard physics subjects such as perturbation and scattering theory, bound-state formation, etc. , but also at the forefrontof current and, most likely, future developments. 3. Over the last 15 yearsor so Green's functions havefound applications not only in condensed matter electronicmotion but in classicalwavepropa- tion in both periodic and random media; photonic and phononic crystals aretheoutcomesofthislineofresearchwhoseunderlyingbasictheoretical principles are summarized in Sect.7.2.4.