  We present a theory of reduction for Courant algebroids as well as Dirac
structures, generalized complex, and generalized K\"ahler structures which
interpolates between holomorphic reduction of complex manifolds and symplectic
reduction. The enhanced symmetry group of a Courant algebroid leads us to
define \emph{extended} actions and a generalized notion of moment map. Key
examples of generalized K\"ahler reduced spaces include new explicit
bi-Hermitian metrics on $\CC P^2$.
