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Electromagnetic metamaterials, artificial media created by subwavelength structuring, are useful for engineering electromagnetic space and controlling light propagation. Such materials exhibit many unusual properties that are rarely or never observed in nature. They can be employed to realize useful functionalities in emerging metadevices based on light. Here, we review hyperbolic metamaterials — one of the most unusual classes of electromagnetic metamaterials. They display hyperbolic (or indefinite) dispersion, which originates from one of the principal components of their electric or magnetic effective tensor having the opposite sign to the other two principal components. Such anisotropic structured materials exhibit distinctive properties, including strong enhancement of spontaneous emission, diverging density of states, negative refraction and enhanced superlensing effects.

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