  We complete the investigation of N=(2,2) supersymmetric nonlinear
sigma-models in the presence of a boundary. We study the full bihermitian
geometry parameterized by chiral, twisted chiral and semi-chiral superfields
and identify the D-brane configurations preserving an N=2 worldsheet
supersymmetry. Combining twisted with semi-chiral superfields leads to a
clearly defined notion of lagrangian and coisotropic branes generalizing
lagrangian and coisotropic A-branes on Kahler manifolds to manifolds which are
not necessarily Kahler (but still bihermitian). Adding chiral fields
complicates the picture and results in hybrid configurations interpolating
between lagrangian/coisotropic branes and branes wrapping around a holomorphic
cycle. Even here the branes can be viewed as coisotropic submanifolds albeit in
a generalized sense. All supersymmetric D-brane configurations are
characterized in the context of generalized complex geometry. Duality
transformations interchanging the various types of superfields while preserving
all supersymmetries are explicitly constructed and provide for a powerful
technique to construct various highly non-trivial D-brane configurations.
Several explicit examples are given.
