Shape plays an important role in many areas of science and engineering. Cross-sectional geometry of the airplane wings determines the aerodynamic behavior of an aircraft; grain sizes and shapes of a material govern its physical properties; hippocampal geometry is known to be a strong marker for mental diseases such as dementia and Alzheimer’s disease; skull shapes can tell so many untold stories about how the human species has evolved throughout the history—just to list only a few examples. In this reason, scientists and engineers are eager to explore the space of shapes, hoping to learn or discover latent patterns and trends behind shape variations. However, quantitative statistical analysis of shapes is, oftentimes, not quite trivial to accomplish due to many practical reasons. For example, quantifying the semantic notion of similarity is a long-standing problem in computational geometry. It can also be astoundingly challenging to establish point-to-point correspondences between two shapes, or, there may not even exist a canonical, statistically consistent representation to describe shapes. To this end, there have been enormous efforts devoted over the last several decades to craft mathematical formulae and algorithms for representing, comparing, and, eventually, understanding different shapes from data. In this talk, I will present a gentle introduction to a manifold-based formalization of statistical shape analysis problems, along with an overview of a few well-known solutions. Especially, I will focus on new solutions leveraging the extended capacity of deep neural networks, which has been disruptively changing the landscape of the statistical shape analysis research over the last few years.