Functional data analysis (FDA) is fast becoming an important research area, due to its broad applications in many branches of science. An essential component in FDA is the registration of points across functional objects. Without proper registration the results are often inferior and difficult to interpret. The current practice in FDA community is to treat registration as a pre-processing step, using off-the-shelf alignment procedures, and follow it up with statistical analysis of the resulting data. In contrast, Elastic FDA is a more comprehensive approach, where one solves for the registration and statistical inferences in a simultaneous fashion. The key idea here is to use Riemannian metrics with appropriate invariance properties, to form objective functions for alignment and to develop statistical models involving functional data. While these elastic metrics are complicated in general, we have developed a family of square-root transformations that map these metrics into simpler Euclidean metrics, thus enabling more standard statistical procedures. Specifically, we have developed techniques for elastic functional PCA and elastic regression models involving functional variables. I will demonstrate this ideas using imaging data in neuroscience where anatomical structures can often be represented as functions (curves or surfaces) on intervals or spheres. Examples of curves include DTI fiber tracts and sulcal folds while examples of surfaces include subcortical structures (hippocampus, thalamus, putamen, etc). Statistical goals here include shape analysis and modeling of these structures and to use their shapes in medical diagnosis.