Metamorphosis between 3D objects is often the transformation between a pair of shapes that have the same topology. This paper presents a new model using Reeb graphs and their contours to create morphing between 3D objects having different topology. The proposed method specifies the correspondence between of the input objects by using the graph isomorphic theory. Then the super Reeb graph, which has the equivalent topological information to the Reeb graphs of the two input objects, is constructed and used to conduct the sequence of the morphing. The evolutions of the topology that occur during the morph can be specified by the transformation of the Reeb graphs and their contours of the input objects. Reeb graph-based modeling allows the users precisely and intuitively control the morph because the topological information of the objects, represented by the structures of the Reeb graphs, is explicit and easy to understand. Moreover, the contours of the Reeb graphs also represent the geometrical information of the objects. The examples of morphing between different topological shapes are demonstrated.