Object Oriented Data Analysis is the statistical analysis of populations of complex objects. This is seen to be particularly relevant in the Big Data era, where it is argued that an even larger scale challenge is Complex Data. Data objects with a geometric structure constitute a particularly active current research area. This is illustrated using a number of examples where data objects naturally lie in manifolds and spaces with a manifold stratification. An overview of the area is given, with careful attention to vectors of angles, i.e. data objects that naturally lie on torus spaces. Prinicpal Nested Submanifolds, which are generalizations of flags, are proposed to provide new analogs of Principal Component Analysis for data visualization. Open problems as to how to weight components in a simultaneous fitting procedure are discussed.