The year 2004 was a remarkable one for the growing ?eld of time-dependent density functional theory (TDDFT). Not only did we celebrate the 40th - niversary of the Hohenberg-Kohn paper, which had laid the foundation for ground-state density functional theory (DFT), but it was also the 20th - niversary of the work by Runge and Gross, establishing a ?rm footing for the time-dependent theory. Because the ?eld has grown to such prominence, and has spread to so many areas of science (from materials to biochemistry), we feel that a volume dedicated to TDDFT is most timely. TDDFT is based on a set of ideas and theorems quite distinct from those governingground-stateDFT,butemployingsimilar techniques.Itisfarmore than just applying ground-state DFT to time-dependent problems, as it - volves its own exact theorems and new and di?erent density functionals. Presently,themostpopularapplicationistheextractionofelectronicexcit- state properties, especially transition frequencies. By applying TDDFT after thegroundstateofamoleculehasbeenfound,wecanexploreandunderstand the complexity of its spectrum, thus providing much more information about the species. TDDFT has a especially strong impact in the photochemistry of biological molecules, where the molecules are too large to be handled by t- ditional quantum chemical methods, and are too complex to be understood with simple empirical frontier orbital theory.

Users Guide.- Abbreviations.- Notation.- Basics of DFT.- Part I Formal Theory.- Part II Approximate Functionals.- Part III Numerical Aspects.- Part IV Applications: Linear Response.- Part V Applications: Beyond Linear Response.- Part VI New Frontiers.