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The technological advances in recent years have produced a wealth of intricate digital imaging data that is analyzed effectively using the principles of shape analysis. Such data often lies on either high-dimensional or infinite-dimensional manifolds. With computing power also now strong enough to handle this data, it is necessary to develop theoretically-sound methodology to perform the analysis in a computationally efficient manner. In this dissertation, we propose approaches of doing so for planar contours and the three-dimensional atomic structures of protein binding sites. First, we adapt Kendall's definition of direct similarity shapes of finite planar configurations to shapes of planar contours under certain regularity conditions and utilize Ziezold's nonparametric view of Frechet mean shapes. The space of direct similarity shapes of regular planar contours is embedded in a space of Hilbert-Schmidt operators in order to obtain the Veronese-Whitney extrinsic mean shape. For computations, it is necessary to use discrete approximations of both the contours and the embedding. For cases when landmarks are not provided, we propose an automated, randomized landmark selection procedure that is useful for contour matching within a population and is consistent with the underlying asymptotic theory. For inference on the extrinsic mean direct similarity shape, we consider a one-sample neighborhood hypothesis test and the use of nonparametric bootstrap to approximate confidence regions. Bandulasiri et al (2008) suggested using extrinsic reflection size-and-shape analysis to study the relationship between the structure and function of protein binding sites. In order to obtain meaningful results for this approach, it is necessary to identify the atoms common to a group of binding sites with similar functions and obtain proper correspondences for these atoms. We explore this problem in depth and propose an algorithm for simultaneously finding the common atoms and their respective correspondences based upon the Iterative Closest Point algorithm. For a benchmark data set, our classification results compare favorably with those of leading established methods. Finally, we discuss current directions in the field of statistics on manifolds, including a computational comparison of intrinsic and extrinsic analysis for various applications and a brief introduction of sample spaces with manifold stratification.