The unprecedented availability of data is revolutionising nearly every field of scientific endeavour. Technological advances let us measure processes and structures in aspects and resolutions that are extraordinary. A considerable challenge is that of scale: the sheer amount of data presents novel challenges and trade-offs between statistical efficiency and computational tractability, and gives rise to new theoretical paradigms. Modern data, however, are not just big. They are often complex, too: further to the challenge of scale, the data often carry intrinsic structure that requires mathematical formalisms escaping the traditional framework of statistics. The sample and parameter spaces may be infinite-dimensional, non-linear, combinatorial, or a combination thereof. This semester will focus on cutting edge statistical questions for functional data structures such as random processes, random measures, and random operators; but also for related discrete data structures such as combinatorial and topological features that are often associated with otherwise intrinsically continuous data. In each of these cases one wishes to extract parsimonious representations, empirical dynamics, and variational features, while quantifying the inherent uncertainty by means of formal statistical procedures. And, any of these frameworks can be further enriched by bringing in corresponding covariates, whether functional or not. The goal of the proposed semester will be to bring together leading experts and new researchers in the field in order to report on recent progress, sketch a landscape of current challenges, and identify important avenues forward. The programme will also explore contact points with researchers whose work has significant interface with functional data analysis, even if it is not typically classified as such, for example researchers from the fields of Bayesian nonparametrics, high dimensional statistics, random networks, numerical analysis, uncertainty quantification, and topological data analysis.