Object oriented data analysis is an important tool in the many disciplines where the data have much richer structure than the usual numbers or vectors. An initial question to ask is: what are the most basic data units? i.e. what are the atoms of the data? We describe an introduction to this topic, where the statistical analysis of object data has a wide variety of applications. An important aspect of the analysis is to reduce the dimension to a small number key features while respecting the geometry of the manifold in which objects lie. Three case studies are given which exemplify the types of issues that are encountered: i) investigating differences between authors in corpus linguistics, ii) describing changes in variability in damaged DNA, iii) testing for geometrical differences in carotid arteries, where patients are at high or low risk of aneurysm. In all three applications the structure of the data manifolds is important, in particular the manifolds of networks, covariance matrices, and unlabeled size-and-shapes.