Complex integration is an important topic in the field of complex analysis. There exist some related criteria between the integration in the real and complex functions. We will start this chapter, by introducing a short review for the line integration in the real domain then followed it by the subject of the present chapter. Theorems, corollaries and other facts will be introduced. After that we now turn to the problem of integrating complex functions. The theory you will learn is elegant, powerful, and a useful tool for physicists and engineers.