A central motivation for the development of mathematical models of shape and methods of shape analysis is the need to understand morphological variation in a variety of contexts. Understanding development, evolution and inheritance of biological traits, quantifying normal and pathological changes in the anatomy of organs and tissues, recognizing objects in images, all involve shape analysis. Many of the problems reside at the interface between geometry, statistics, and pattern analysis. In this talk, we will discuss: (i) interpolation techniques for regularizing shapes to make them amenable to analysis; (ii) construction of shape spaces and metrics that provide a framework for modeling shape variation. We also will illustrate the methods with applications to biology and medical imaging.