  We take a fresh look at the relation between generalised K\"ahler geometry
and $N=(2,2)$ supersymmetric sigma models in two dimensions formulated in terms
of $(2,2)$ superfields. Dual formulations in terms of different kinds of
superfield are combined to give a formulation with a doubled target space and
both the original superfield and the dual superfield. For K\"ahler geometry, we
show that this doubled geometry is Donaldson's deformation of the holomorphic
cotangent bundle of the original K\"ahler manifold. This doubled formulation
gives an elegant geometric reformulation of the equations of motion. We
interpret the equations of motion as the intersection of two Lagrangian
submanifolds (or of a Lagrangian submanifold with an isotropic one) in the
infinite dimensional symplectic supermanifold which is the analogue of phase
space. We then consider further extensions of this formalism, including one in
which the geometry is quadrupled, and discuss their geometry.
