In this thesis, we introduce a comprehensive framework for the analysis of statistical samples that are functional data with non-trivial geometry. Geometry can interplay with functional data in different forms. The most general setting considered here is that of functional data supported on random non-linear smooth manifolds. This is a situation often encountered in neuroimaging, where modern imaging modalities are now able to produce structural brain representations coupled with functional information. Practitioners have commonly approached the analysis of such data with a two step approach. In the first step the manifolds are registered to a template and in the second step the functional information is analyzed on the template ignoring the registration step. The separation of the two steps precludes studies aimed at understanding how geometric variations relate to functional variations. On the other hand, functional data analysis has mostly developed tools for simplified settings, such as one-dimensional functional samples, limiting their applicability to real data. We formulate a model which is able to jointly represent geometric and functional variations. In this setting, modeling functional information requires the formulation of models able to incorporate structural information on the geometry of the underlying domains, with the aim of mitigating the curse of dimensionality. This is achieved by adopting regularized models involving differential operator penalties. Modeling random smooth manifolds requires the formulation of models constrained to produce sensible' shapes, e.g. not self-intersecting. This is achieved by means of diffeomorphic flows. The proposed models have been applied to real data to perform studies able to relate structural changes to functional changes, and specifically, to study associations between brain shape and cerebral cortex thickness. We can also deal with more complex functional samples, themselves constrained to lie in a non-linear subspace. This is for instance the case of covariance operators, describing brain connectivity, which are symmetric and positive semi-definite operators. Thanks to the proposed models, we are able to model connectivity as an object' and study its variations in time or across individuals. We also consider further extensions of this framework to the inverse problems setting, which is the setting where each sample is a latent object, and only indirect measurements are available.