A relatively new method of materials structure analysis — positron annihilation spectroscopy (PAS) — is reviewed. Measurements of positron lifetimes, the determination of positron 3γ- and 2γ-annihilation probabilities, and an investigation of the effects of different external factors on the fundamental characteristics of annihilation constitute the basis for this promising method. The ways in which the positron annihilation process operates in ionic crystals, semiconductors, metals and some condensed matter systems are analyzed. The scope of PAS is described and its prospects for the study of the electronic and defect structures are discussed. The applications of positron annihilation spectroscopy in radiation physics and chemistry of various substances as well as in physics and chemistry of solutions are exemplified.