For any two closed orientable surfaces, there is a map of nonzero degree from one to another. This result is based on the fact that the orientable closed surface are classified completely. There is no such a classification in dimension 3. In this paper, all closed aspherical 3-manifolds are divided into eight classes. A complete answer of the existences of nonzero degree maps between 3-manifolds in the different classes is obtained in a reasonable sense. The existence of nonzero degree maps between 3-manifolds in the same class are also discussed. There are several results of independent interest. It is noted that, the existence of a map of nonzero degree between two 3-manifolds is a very restrictive condition posed on those two 3-manifolds, which is much different from the case of dimension 2. The shape of our work is certainly influenced by Thurston. However, both the statements and the proofs of our theorems do not involve the Thurston's geometries. We understand that if we state the results in the category of Haken manifolds and Thurston's geometrical manifolds and invoke the G r o m o v Norm of Hyperbol ic 3-manifold (see Chap. 6 of r T l ] and Chap. 4 of [-Gro]), the proof of some result in the paper can be simplified. We present our work in the way below for two reasons: (a) we prefer to prove all results in a uniform elementary way in the sense of using only the fundamental groups and the incompressible surfaces of 3-manifolds; (b)since the Thurston's most famous conjecture that every atoroidal closed irreducible 3-manifold has a hyperbolic structure is still open, we get the results in a theoretically wider category. There are 5 sections after the introduction. All manifolds and maps under consideration are piece-wise linear. (There is nothing lost under this assumption, see Chap. 1 of [Hi.) In the first four sections of these five, all 3-manifolds are orientable and irreducible. Section 1 states the theorems proved in the paper. Section 2 provides results which are used to prove the theorems. Section 3 and Sect. 4 are the proofs of the theorems. Various examples are given in Sect. 4.