Abstract: The irreducible (real) Clifford modules play a very important role in the theory of Dirac operators. The obstruction for the existence of such a module in real and complex case is different. For the real one, vector bundle should have vanishing second Stiefel-Whitney class, however, in the complex case the second Stiefel-Whitney class only needs to be mod 2 reduction of an integral class, equivalently, the vector bundle admits a $spin^c$ structure. In this talk we will examine the obstruction for the existence of $spin^c$ structure, then the construction of complex spinor bundle and the $spin^c$ connection on it and finally Dirac operator on the spinor fields will be discussed.