This review of the properties of flexible magnetic filaments covers the problems of their buckling under the action of a magnetic field. The dependence of the buckling on the magnetic properties of the filaments is considered. It is shown that these filaments may be used as artificial analogues of structures with important roles in the living world: cilia, elastic tails of self-propelling microorganisms etc. The main methods used to study the relevant phenomena are considered: models, numerical simulation algorithms, and the results of linear stability analysis. Special emphasis is placed on the possible uses of the flexible magnetic filaments as model systems to study the dynamics of magnetic fields using magnetometers based on optical detection of the magnetic resonance of NV-centers in diamonds.