Shapes are all around us, and relationships between them are understood intuitively. These relationships are often analyzed by means of using different similarity metrics. These similarity metrics provide understanding about the salient features of a set of shapes, providing pointers to what the similarities on a set of shapes are, and how they are different from other shapes. With increased availability of three-dimensional data, the need for understanding different shape modalities and unifying information obtained from them becomes crucial. This thesis delves deep into how shape geometry and shape structure can be used as similarity metrics to analyze shape relationships. This is followed by real-world applications of obtaining these similarity-based shape relationships. The similarity of shape geometry is shown to be useful in obtaining shape correspondences on pairs of shapes and shape collections. Structural similarity on shape collections, provide different kinds of structural priors, which are useful in completing partial shapes. This thesis work has pioneered shape correspondences between nonisometric shapes. While shape matching on isometric pairs of shapes is a well-studied problem, most real-world shape pairs and collections are nonisometric in nature, and this is a crucial problem to solve. An novel consideration that we introduce on computing maps on shape collections, is the notion of modularity. While consistency is an important consideration that is already in use, coupling it with modularity makes the correspondence computation process much quicker. While these two problems involve using shape geometry as a cornerstone to solve real world correspondences, we also shape structure as an important similarity metric, and use it to complete real scans. Here, we discuss two different paradigms in shape completion. We introduce structure-aware shape templates, that can be useful in performing shape completion. To this effect, we provide a template-based optimization approach, which is coupled with a multi-view CNN to infer the structure of complete and partial point clouds. This is shown to be useful in partial scan completion, scene completion and segmentation tasks. In addition to this retrieval-based shape completion technique, we also provide a surface reconstruction technique on point clouds that utilizes topological structure of the shape. This surface completion technique is unique in the fact that it provides a handle on solving the hard combinatorial topology-aware optimization problem. As these techniques are more general in nature, they can be useful in spheres which are even removed from shape similarity understanding. This is discussed in the concluding segment of the thesis, along with directions for future research.