We review the time-dependent density functional theory (TDDFT) and its use to investigate excited states of nanostructures. These excited states are routinely probed using electromagnetic fields. In this case, two different regimes are usually distinguished: (i) If the electromagnetic field is “weak”— as in optical absorption of light—it is sufficient to treat the field within linear response theory; (ii) Otherwise, nonlinear effects are important, and one has to resort to the full solution of the timedependent Kohn-Sham equations. This latter regime is of paramount relevance in the emerging field of research with intense and ultrashort laser pulses. This review is divided into two parts: First we give a brief overview of the theoretical foundations of the theory, both in the linear and non-linear regimes, with special emphasis on the problem of the choice of the exchange-correlation functional. Then we present a sample of applications of TDDFT to systems ranging from atoms to clusters and to large biomolecules. Although most of these applications are in the linear regime, we show a few examples of non-linear phenomena, such as the photo-induced dissociation of molecules. Many of these applications have been performed with the recently developed code octopus (http://www.tddft.org/programs/octopus).