Abstract: For any manifold $M$ with a spin structure $P_{spin}(X)$, we can define a canonical first order operator, known as the Dirac operator, acting on a spinor bundle. The spinor bundle is a bundle associated to the spin structure and the representation of spin group, coming from an irreducible representation of the Clifford algebra. The existence of an irreducible real Clifford module is equivalent to the existence of a spin structure on the manifold; however, we can always construct a Dirac bundle, which is (not necessarily irreducible) a $Cl(X)$-module with compatible metric and connection. Similarly, we may define a differential operator known as the generalized Dirac operator. In this talk, after some general discussion about Clifford bundles and Dirac bundles, we will focus on two important examples of the generalized Dirac operator: the de Rham complex and, in the $4k$-dimensional case, the signature complex.