The measurement of the time-dependent spectrum of non-stationary radiation is of interest, for instance, in the field of ultrashort-time spectroscopy using picosecond lasers. The incident non-stationary radiation is characterized by the time-dependent input spectrum. The electric field strength of the radiation in the exit plane of a grating spectrometer and of a prism spectrometer is analysed in terms of its dependence on time and on the spectral coordinate. The action of the grating spectrometer is regarded as a special case of diffraction at an arbitrary diaphragm. The intensity registered by a square-law detector as a function of time and the spectral coordinate (output spectrum) is calculated as a function of the input spectrum. The input spectrum can be determined from the output spectrum by mathematical operations for cases important in practise (very short light-pulses, intrinsically stationary radiation). Moreover, the conditions are investigated under which the output spectrum resembles the input spectrum, taking into account the influence of noise. Two examples of intrinsically stationary radiation (bandwidth-limited pulse, fluorescence of a two-level system) are studied in more detail. For experimental verification the ouput spectrum of a single ultrashort light pulse has been measured under the condition of different spectrometer time constants.