${\cal N}=2$ Supergravity in $D=4,5,6$ Dimensions

An overview of matter-coupled ${\cal N}=2$ supergravity theories with 8 real supercharges, in 4,5 and 6 dimensions is given. The construction of the theories by superconformal methods is explained from basic principles. Special geometry is obtained and characterized. The relation between the theories in those dimensions is discussed. This leads to the concepts of very special geometry and quaternionic-Kähler manifolds, whose structures are explained.

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