  We introduce the notion of a special complex manifold: a complex manifold
(M,J) with a flat torsionfree connection \nabla such that (\nabla J) is
symmetric. A special symplectic manifold is then defined as a special complex
manifold together with a \nabla-parallel symplectic form \omega . This
generalises Freed's definition of (affine) special K\"ahler manifolds. We also
define projective versions of all these geometries. Our main result is an
extrinsic realisation of all simply connected (affine or projective) special
complex, symplectic and K\"ahler manifolds. We prove that the above three types
of special geometry are completely solvable, in the sense that they are locally
defined by free holomorphic data. In fact, any special complex manifold is
locally realised as the image of a holomorphic 1-form \alpha : C^n \to T^* C^n.
Such a realisation induces a canonical \nabla-parallel symplectic structure on
M and any special symplectic manifold is locally obtained this way. Special
K\"ahler manifolds are realised as complex Lagrangian submanifolds and
correspond to closed forms \alpha. Finally, we discuss the natural geometric
structures on the cotangent bundle of a special symplectic manifold, which
generalise the hyper-K\"ahler structure on the cotangent bundle of a special
K\"ahler manifold.
