In this first chapter, the reader is acquainted with the concept of space tether systems and the respective areas of their application. In chronological order, we review experiments in which space tethers were used. The chapter provides an overview of existing materials used for tether construction, and there is an extensive review of the literature devoted to the problem. Types of mathematical models, the various modes of space tether systems motion, questions of stability and control, the task of spin-up of the system, and some poorly investigated topics revealed as a result of the scientific literature analysis are considered.