Generalized complex geometry was first introduced by N. Hitchin and then further developed by Gualtieri as a simultaneous generalization of both symplectic and complex geometries. A notion of generalized moment map and Hamiltonian actions in generalized complex geometry were introduced by Tolman and the speaker a while ago. In this talk, we first explain how the quotient construction can be used to produce explicit examples of bi-Hermitian structures; we then discuss the Morse-Bott theory behind the geometry of generalized moment maps. In particular, for Hamiltonian torus actions on compact generalized complex manifolds this leads to Kirwan surjectivity results for both the usual equivariant cohomology and the twisted equivariant cohomology theory.