The primary aim of experimental nuclear astrophysics is to determine the rates of nuclear reactions taking place in stars in various astrophysical conditions. These reaction rates are important ingredient for understanding the elemental abundance distribution in our solar system and the galaxy. The reaction rates are determined from the cross sections which need to be measured at energies as close to the astrophysically relevant ones as possible. In many cases the final nucleus of an astrophysically important reaction is radioactive which allows the cross section to be determined based on the off-line measurement of the number of produced isotopes. In general, this technique is referred to as the activation method, which often has substantial advantages over in-beam particle- or gamma-detection measurements. In this paper the activation method is reviewed from the viewpoint of nuclear astrophysics. Important aspects of the activation method are given through several reaction studies for charged particle, neutron and gamma-induced reactions. Various techniques for the measurement of the produced activity are detailed. As a special case of activation, the technique of Accelerator Mass Spectrometry in cross section measurements is also reviewed.