EXAFS spectroscopy is a new method of investigating materials which allows one to determine structural parameters of the local environment of atoms with some specified $\mathbf{Z}$ by studying their x-ray spectra. Among these parameters are the interatomic spacings, coordination numbers and amplitudes of thermal oscillations. It is not necessary for long-range order to be present in the sample under investigation. Depending on the way this technique for obtaining the spectrum is applied, one can analyze the local environment of atoms located either within the sample volume or at its surface. We investigate the physical phenomena on which the method is based, the mathematical techniques used to process the experimental data, and various methods of recording the spectra. We present a series of examples in which EXAFS spectroscopy is used to study superionic conductors, compounds with intermediate valence, biological molecules, solid solutions, catalysts, surface layers and intercalated compounds.