This book provides a systematic and unifying treatment of the spectral analysis of matrix sequences collecting the state of the art of the research in this field and providing new advanced results which are interesting both from the theoretical and from the algorithmic point of view. The first part concerns the description and analysis of the available tools, together with their generalization including the development of new effective tools of analysis. The basic concept of "approximating class of sequences" is introduced and analyzed in depth. Some new results of great interest have been...