In this article we aim to highlight the problems related to the structure and stability of the comparatively thin current sheets that were relatively recently discovered by space missions in the magnetospheres of the Earth and planets, as well as in the solar wind. These magnetoplasma structures are universal in collisionless cosmic plasmas and can play a key role in the processes of storage and release of energy in the space environment. The development of a self-consistent theory for these sheets in the Earth's magnetosphere, where they were first discovered, has a long and dramatic history. Solution of the problem of the thin current sheet structure and stability become possible in the framework of a kinetic quasi-adiabatic approach required to explain their embedding and metastability properties. It was found that the structure and stability of current structures are completely determined by the nonlinear dynamics of plasma particles. Theoretical models have been developed to predict many properties of these structures and interpret many experimental observations in planetary magnetospheres and the heliosphere.