High-T_c superconductors in small magnetic fields directed away from the crystal symmetry axes have been found to exhibit inhomogeneous chains of flux lines (vortices), in contrast to the usual regular triangular flux-line lattice. We review the experimental observations of these chains, and summarize the theoretical background that explains their appearance. We treat separately two classes of chains: those that appear in superconductors with moderate anisotropy due to an attractive part of the interaction between tilted flux lines, and those with high anisotropy where the tilted magnetic flux is created by two independent and perpendicular crossing lattices. In the second case it is the indirect attraction between a flux line along the layers (Josephson vortex) and a flux line perpendicular to the layers (pancake vortex stack) that leads to the formation of chains of the pancake vortex stacks. This complex system contains a rich variety of phenomena, with several different equilibrium phases, and an extraordinary dynamic interplay between the two sets of crossing vortices. We compare the theoretical predictions of these phenomena with the experimental observations made to date. We also contrast the different techniques used to make these observations. While it is clear that this system forms a wonderful playground for probing the formation of structures with competing interactions, we conclude that there are important practical implications of the vortex chains that appear in highly anisotropic superconductors.
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