Functional Data Analysis has shaped the way we perform statistical analysis on random samples that are functions on the real line.
However, thanks to recent developments in the field we are now able to redefine the way we do statistical analysis on much more complex objects. In this talk, I will introduce a comprehensive framework for the analysis of functional data whose domain is a manifold and the domain itself is subject to variability from sample to sample. I will also cover further extensions of this framework to the inverse problem setting, where the samples themselves are latent objects, and only indirectly measured signals are available. To illustrate the proposed ideas, I will show several applications of the proposed models; these will be mainly related to medical image analysis.