A physically-based approach is presented to fitting complex 3D shapes using a novel class of dynamic models. These models can deform both locally and globally. The authors formulate deformable superquadrics which incorporate the global shape parameters of a conventional superellipsoid with the local degrees of freedom of a spline. The local/global representational power of a deformable superquadric simultaneously satisfies the conflicting requirements of shape reconstruction and shape recognition. The model's six global deformational degrees of freedom capture gross shape features from visual data and provide salient part descriptors for efficient indexing into a database of stored models. Model fitting experiments involving 2D monocular image data and 3D range data are reported.