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Existing methods of morphing 3D meshes are often limited to cases in which 3D input meshes to be morphed are topologically equivalent. This paper presents a new method for morphing 3D meshes having different surface topological types. The most significant feature of the method is that it allows explicit control of topological transitions that occur during the morph. Transitions of topological types are specified by means of a compact formalism that resulted from a rigorous examination of singularities of 4D hypersurfaces and embeddings of meshes in 3D space. Using the formalism, every plausible path of topological transitions can be classified into a small set of cases. In order to guide a topological transition during the morph, our method employs a key-frame that binds two distinct surface topological types. The key-frame consists of a pair of "faces", each of which is homeomorphic to one of the source (input) 3D meshes. Interpolating the source meshes and the key-frame by using a tetrahedral 4D mesh and then intersecting the interpolating mesh with another 4D hypersurface creates a morphed 3D mesh. We demonstrate the power of our methodology by using several examples of topology transcending morphing.