Geometric analysis of shapes plays an important role in the way the visual world is perceived by modern computers. To this end, low-level geometric features provide most obvious and important cues towards understanding the visual scene. A novel intrinsic geometric surface descriptor, termed as the Geodesic Field Estimate (GFE) is proposed. Also proposed is a parallel algorithm, well suited for implementation on Graphics Processing Units, for efficient computation of the shortest geodesic paths. Another low level geometric descriptor, termed as the Biharmonic Density Estimate, is proposed to provide an intrinsic geometric scale space signature for multiscale surface feature-based representation of deformable 3D shapes. The computer vision and graphics communities rely on mid-level geometric understanding as well to analyze a scene. Symmetry detection and partial shape matching play an important role as mid-level cues. A comprehensive framework for detection and characterization of partial intrinsic symmetry over 3D shapes is proposed. To identify prominent overlapping symmetric regions, the proposed framework is decoupled into Correspondence Space Voting followed by Transformation Space Mapping procedure. Moreover, a novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric embedding is also proposed. The ultimate goal of geometric shape analysis is to resolve high level applications of modern world. This dissertation has focused on three different application scenarios. In the first scenario, a novel approach for the analysis of the non-rigid Left Ventricular (LV) endocardial surface from Multi-Detector CT images, using a generalized isometry-invariant Bag-of-Features (BoF) descriptor, is proposed and implemented. In the second scenario, the geometric regularity and variability of the cortical surface fold patterns at the 358 Dense Individualized and Common Connectivity-based Cortical Landmarks (DICCCOL) sites are quantitatively analyzed using Geodesic Context Histogram, a histogram constructed using GFE values in the spatial neighborhood of a surface point. In the third and final application scenario, we formulate a partial shape matching based technique, that can analyze the structure of the geometric shapes of the images and match them in a concise and meaningful manner directly, rather than relying on metadata to solve the problem of Content Based Image Retrieval.