  Special Kahler manifolds are defined by coupling of vector multiplets to
$N=2$ supergravity. The coupling in rigid supersymmetry exhibits similar
features. These models contain $n$ vectors in rigid supersymmetry and $n+1$ in
supergravity, and $n$ complex scalars. Apart from exceptional cases they are
defined by a holomorphic function of the scalars. For supergravity this
function is homogeneous of second degree in an $(n+1)$-dimensional projective
space. Another formulation exists which does not start from this function, but
from a symplectic $(2n)$- or $(2n+2)$-dimensional complex space. Symplectic
transformations lead either to isometries on the manifold or to symplectic
reparametrizations. Finally we touch on the connection with special
quaternionic and very special real manifolds, and the classification of
homogeneous special manifolds.
