This text is concerned with the relationship between spectral decomposition of operators, the functional calculi that operators admit, and Banach space structure. The deep connection between the first two of these concepts has long been known. The aim of this text is to provide an introduction to the modern techniques of AC- operators and prespectral operators in spectral theory. Topics include the basic results of AC- operators and prespectral operators on Banach spaces and the application of spectral theory on c*-algebras. Much of the subject matter, particularly with respect to AC- operators, prespectral operators, and c*-algebras, is comprised of new results.