Nigel Hitchin introduces a geometrical structure on even-dimensional manifolds that generalizes both Calabi-Yau manifolds and symplectic manifolds. This structure, called a *generalized complex manifold*, is defined using the *Courant bracket* and *B-field*. The Courant bracket is a generalization of the Lie bracket to sections of the bundle $T \oplus T^*$, while the B-field is a closed 2-form that acts on the structure. Generalized complex manifolds can be either odd or even, and they transform naturally under diffeomorphisms and the action of closed 2-forms. In six dimensions, Hitchin characterizes these structures as critical points of a natural variational problem on closed forms and proves that a local moduli space is provided by an open set in either the odd or even cohomology. He also discusses the role of the B-field in interpolating between symplectic and Calabi-Yau structures. The paper includes detailed definitions, examples, and proofs, including the construction of a pseudo-Kähler structure on the moduli space and the use of gerbes with connection to twist the structure. Hitchin's work is motivated by the need to characterize special geometry in low dimensions and has implications for string theory. The paper covers the Courant bracket, spinors, pure spinors, and the definition of generalized complex and generalized Calabi-Yau manifolds. It also explores the geometry of the spin representation and the variational problem, including the volume functional and the Hessian of the functional at critical points. The paper concludes with a discussion of the moduli space and the conditions under which it is non-degenerate.Nigel Hitchin introduces a geometrical structure on even-dimensional manifolds that generalizes both Calabi-Yau manifolds and symplectic manifolds. This structure, called a *generalized complex manifold*, is defined using the *Courant bracket* and *B-field*. The Courant bracket is a generalization of the Lie bracket to sections of the bundle $T \oplus T^*$, while the B-field is a closed 2-form that acts on the structure. Generalized complex manifolds can be either odd or even, and they transform naturally under diffeomorphisms and the action of closed 2-forms. In six dimensions, Hitchin characterizes these structures as critical points of a natural variational problem on closed forms and proves that a local moduli space is provided by an open set in either the odd or even cohomology. He also discusses the role of the B-field in interpolating between symplectic and Calabi-Yau structures. The paper includes detailed definitions, examples, and proofs, including the construction of a pseudo-Kähler structure on the moduli space and the use of gerbes with connection to twist the structure. Hitchin's work is motivated by the need to characterize special geometry in low dimensions and has implications for string theory. The paper covers the Courant bracket, spinors, pure spinors, and the definition of generalized complex and generalized Calabi-Yau manifolds. It also explores the geometry of the spin representation and the variational problem, including the volume functional and the Hessian of the functional at critical points. The paper concludes with a discussion of the moduli space and the conditions under which it is non-degenerate.