This paper presents a new general approach to blend 2D shapes with different topologies. All possible topolog ical evolutions are classified into three types by attaching three different topological cells. This formalism is resulted from Morse theory on the behavior of the 3D surface around a non-degenerate critical point. Also we incorporate degenerate topological evolutions into our framework which produce more attractive morphing effects. The user controls the morph by specifying the types of topological evolutions as well as the feature correspondences between the source and target shapes.Some techniques are also provided to control the vertex path during the morphing process. The amount of user input required to produce a morph is directly proportional to the amount of control the user wishes to impose on the process.The user may allow the system to automatically generate the morph as well. Our approaches are totally geometric based and are easy and fast enough in fully interactive time. Many experimental results show the applicability and flexibility of our approaches.