Time-dependent density-functional theory (TDDFT) is a quantum mechanical framework which describes the dynamics of interacting electronic many-body systems formally exactly and in a computationally efficient manner. This book presents the concepts of TDDFT at the graduate level. An overview is given of the current state of the art of the field, with many applications in physics, (bio)chemistry, and materials science. The book begins with a self-contained review of ground-state density-functional theory, followed by four main parts. The first part presents the formal framework of TDDFT, beginning with the basic existence theorems of Runge, Gross, and van Leeuwen. The time-dependent Kohn-Sham scheme and the properties of the time-dependent exchange-correlation potential are then discussed in detail. The second part deals with the TDDFT formulation of linear-response theory, which can be used to calculate electronic excitation energies in principle exactly using the Casida equation. An overview is given of the performance of TDDFT for atomic and molecular excitation energies, with many examples. Special attention is paid to challenging cases such as double or charge-transfer excitations. The third part of the book focuses on further developments, including time-dependent current-density-functional theory, the optimized effective potential, and the connection to many-body theory. This part also deals with excitations in extended systems, such as plasmons and excitons. Several special TDDFT topics are discussed in part four: van der Waals interactions, nanoscale transport, strong-field phenomena, and coupled electron-nuclear motion and molecular dynamics. Additional resources are provided in several appendices.