We present a deformation-driven approach to topology-varying 3D shape correspondence. In this paradigm, the best correspondence between two shapes is the one which incurs a minimum-energy, possibly topology-varying, deformation that transforms one shape to conform to the other while respecting the correspondence. Our deformation model, called GeoTopo transform, allows both geometric and topological operations such as part split, duplication, and merging, leading to fine-grained and piecewise continuous correspondence results. The key ingredient of our correspondence scheme is a deformation energy which penalizes geometric distortion, encourages structure preservation, and at the same time, allows topology changes. This is accomplished by connecting shape parts using structural rods, which behave like virtual springs but encode energies arising from geometric, structural, and topological shape variations. Driven by the deformation energy, an optimal shape correspondence is obtained via a pruned beam search. We demonstrate our deformation-driven approach on a variety of man-made models with rich geometric and topological variations.