A review is given of theoretical concepts and experimental data concerning the spin-Peierls transition in a one-dimensional spin system with antiferromagnetic exchange interaction (an analog of Peierls instability of a one-dimensional metal). Analysis of experimental data confirms the existence of the spin-Peierls transition in TTF-CuBDT, TTF-AuBDT, and MEM(TCNQ)$_2$ crystals. The magnitude of the spin-phonon interaction in crystals undergoing the spin-Peierls transition at low temperatures is discussed together with the role of fluctuations in transitions of this type. The influence of magnetic fields on spin-Peierls transitions is examined.