An overview of experimental and theoretical studies of the characteristics of maglev systems using high-temperature superconductors (HTSC) is presented. Materials used in maglev technologies, namely bulk superconductors and HTSC tape composites, are considered. The main experimental data obtained on both bulk and tape superconductors assembled in stacks of various configurations are demonstrated. The factors influencing the magneto-force characteristics are analyzed: geometric parameters, the influence of external alternating magnetic fields, temperatures, relaxation phenomena. A significant part of the review is devoted to the description of various methods for calculating maglev systems, including those based on stacks of HTSC composites. The features of thermal processes in maglev systems with cryocooler and nitrogen cooling are considered. General recommendations for the creation of optimal maglev systems based on tape HTSC composites are given.