In this chapter we establish a (local) geometric intersection theory for cycles in complex manifolds. We use an axiomatic method and prove that there exists a unique intersection theory satisfying four axioms, stated in Sect. 7.1.3, saying roughly that the intersection is compatible with restrictions to open subsets and cycle addition, the intersection of two transversal submanifolds is simply the intersection submanifold and, for two analytic families of cycles such that every cycle in one of the families intersects properly every cycle in the other family, the family of intersection cycles is also analytic. For the proof we use, in an essential way, analytic fundamental classes of cycles as well as relative fundamental classes of analytic families of cycles and their cup products.