The great challenge in computer graphics and geometric-aided design is to devise computationally efficient and optimal algorithms for estimating 3D models contaminated by noise and preserving their geometrical and topological structure. Motivated by the good performance of anisotropic diffusion in 2D image processing, we propose in this thesis a vertex-based nonlinear flow for 3D mesh smoothing by solving a discrete partial differential equation. The core idea behind our proposed technique is to use geometric insight in helping construct an efficient and fast 3D mesh smoothing strategy to fully preserve the geometric structure of the data. Illustrating experimental results demonstrate a much improved performance of the proposed approach in comparison with existing methods currently used in 3D mesh smoothing.  The major part of this thesis is devoted to a joint exploitation of geometry and topology to design new skeletal graph representation of 3D shapes in the Morse-theoretic framework. We present a multiresolution skeletal graph for topological coding of 3D shapes. The proposed skeletonization algorithm encodes a 3D object into a topological graph using a normalized mixture distance function. The approach is accurate, invariant to Euclidean transformations, computationally efficient, and preserves topology. Experimental results demonstrate the potential of the proposed topological graph which may be used as a shape signature for 3D object matching and retrieval, and also for skeletal animation