The problems along the road to achieving an extreme compression of plasma in high-current channels are analyzed. The present ideas regarding the physical nature of the equilibrium of a plasma in a pinch system are outlined. The problem of the macroscopic stability of a relativistic pinch is discussed. Solving this problem requires deriving a relativistic two-fluid electromagnetic hydrodynamics. The dynamics of the contraction of pinches, in whose terminal stage radiation plays a particularly important role, is analyzed. The system can radiate off more energy than it acquires by heat evolution, and if it does the result should be a collapse. The practical significance of achieving electromagnetic collapse goes beyond the fusion problem, and involves such fields as the collective acceleration of charges, the production of ``cumulative'' jets, and the development of intense sources of induced radiation over a broad spectral range, from microwaves to $\gamma$ rays. For physicists, pinches represent a unique opportunity for producing and studying in the laboratory matter in an extreme state such as nature furnishes only deep in the interiors of stars.