  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-K\"ahler manifolds, whose structures are explained.
