In the presence of increasingly massive and heterogeneous data, the statistical modeling of distributional observations plays a key role. Choosing the ‘right’ embedding space for these data is of paramount importance for their statistical processing, to account for their nature and inherent constraints. The Bayes space theory is a natural embedding space for distributional data and was successfully applied in varied settings. In this presentation, I will discuss the state-of-the-art approached for the modelling, analysis, and prediction of distributional data. I will embrace the viewpoint of object-oriented data analysis (OODA), a system of ideas for the analysis of complex data when their dimensionality and inherent properties need careful treatment (such as in the case of distributional constraints). All the theoretical developments will be illustrated through their application on real data, highlighting the intrinsic challenges of a statistical analysis which follows the Bayes spaces approach. I will finally discuss a recent approach, based on alpha-transformations, which can be used to tackle specific limitations of the Bayes space approach, e.g., the presence of zeros in density data.