Basics -- Basics -- Formal Theory -- Beyond the Runge-Gross Theorem -- to the Keldysh Formalism -- Initial-State Dependence and Memory -- Current Density Functional Theory -- Multicomponent Density-Functional Theory -- Intermolecular Forces and Generalized Response Functions in Liouville Space -- Approximate Functionals -- Time-Dependent Deformation Approximation -- Exact-Exchange Methods and Perturbation Theory along the Adiabatic Connection -- Approximate Functionals from Many-Body Perturbation Theory -- Exact Conditions -- Numerical Aspects -- Propagators for the Time-Dependent Kohn-Sham Equations -- Solution of the Linear-Response Equations in a Basis Set -- Excited-State Dynamics in Finite Systems and Biomolecules -- Time Versus Frequency Space Techniques -- Applications: Linear Response -- Linear-Response Time-Dependent Density Functional Theory for Open-Shell Molecules -- Atoms and Clusters -- Semiconductor Nanostructures -- Solids from Time-Dependent Current DFT -- Optical Properties of Solids and Nanostructures from a Many-Body f xc Kernel -- Linear Response Calculations for Polymers -- Biochromophores -- Excited States and Photochemistry -- Applications: Beyond Linear Response -- Atoms and Molecules in Strong Laser Fields -- Highlights and Challenges in Strong-Field Atomic and Molecular Processes -- Cluster Dynamics in Strong Laser Fields -- Excited-State Dynamics in Extended Systems -- New Frontiers -- Back to the Ground-State: Electron Gas -- The Exchange-Correlation Potential in the Adiabatic-Connection Fluctuation-Dissipation Framework -- Dispersion (Van Der Waals) Forces and TDDFT -- Kohn-Sham Master Equation Approach to Transport Through Single Molecules -- Time-Dependent Transport Through Single Molecules: Nonequilibrium Green’s Functions -- Scattering Amplitudes.

Time-dependent density functional theory (TDDFT) is based on a set of ideas and theorems quite distinct from those governing ground-state DFT, but emphasizing similar techniques. Today, the use of TDDFT is rapidly growing in many areas of physics, chemistry and materials sciences where direct solution of the Schrödinger equation is too demanding. This is the first comprehensive, textbook-style introduction to the relevant basics and techniques.