This communication explores Object-Oriented Spatial Statistics (O2S2) as a framework for addressing the challenges of analyzing spatially dependent, complex data structures, such as curves, surfaces, or distributions. By highlighting recent advancements in the field, we will demonstrate how geometric principles can be utilized to model spatially distributed functional data, while accounting for non-stationary behaviors and non-Euclidean spatial domains. A particular emphasis will be given to carefully selecting the embedding space for the analysis, particularly when data exhibit specific features and constraints – a key example being the case of probability density functions, which can be effectively modeled within Bayes spaces. Despite notable progress, challenges remain to build effective estimation methods for the covariance structure of functional fields under non-stationary or non-Euclidean settings, with these challenges being only partially alleviated by using integrated notions of spatial dependence, such as the trace-covariogram and trace-variogram.