Magnetic vortices are typically the ground states in geometrically confined ferromagnets with small magnetocrystalline anisotropy. In this article I review static and dynamic properties of the magnetic vortex state in small particles with nanoscale thickness and sub-micron and micron lateral sizes (magnetic dots). Magnetic dots made of soft magnetic material shaped as flat circular and elliptic cylinders are considered. Such mesoscopic dots undergo magnetization reversal through successive nucleation, displacement and annihilation of magnetic vortices. The reversal process depends on the stability of different possible zero-field magnetization configurations with respect to the dot geometrical parameters and application of an external magnetic field. The interdot magnetostatic interaction plays an important role in magnetization reversal for dot arrays with a small dot-to-dot distance, leading to decreases in the vortex nucleation and annihilation fields. Magnetic vortices reveal rich, non-trivial dynamical properties due to existance of the vortex core bearing topological charges. The vortex ground state magnetization distribution leads to a considerable modification of the nature of spin excitations in comparison to those in the uniformly magnetized state. A magnetic vortex confined in a magnetically soft ferromagnet with micron-sized lateral dimensions possesses a characteristic dynamic excitation known as a translational mode that corresponds to spiral-like precession of the vortex core around its equilibrium position. The translation motions of coupled vortices are considered. There are, above the vortex translation mode eigenfrequencies, several dynamic magnetization eigenmodes localized outside the vortex core whose frequencies are determined principally by dynamic demagnetizing fields appearing due to restricted dot geometry. The vortex excitation modes are classified as translation modes and radially or azimuthally symmetric spin waves over the vortex ground state. Studying the spin eigenmodes in such systems provides valuable information to relate the particle dynamical response to geometrical parameters. Unresolved problems are identified to attract attention of researchers working in the area of nanomagnetism.
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