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Cosmic demographics—the statistical study of populations of astrophysical objects—has long relied on tools from multivariate statistics for analyzing data comprising fixed-length vectors of properties of objects, as might be compiled in a tabular astronomical catalog (say, with sky coordinates, and brightness measurements in a fixed number of spectral passbands). But beginning with the emergence of automated digital sky surveys, ca. 2000, astronomers began producing large collections of data with more complex structures: light curves (brightness time series) and spectra (brightness vs. wavelength). These comprise what statisticians call functional data—measurements of populations of functions. Upcoming automated sky surveys will soon provide astronomers with a flood of functional data. New methods are needed to accurately and optimally analyze large ensembles of light curves and spectra, accumulating information both along individual measured functions and across a population of such functions. Functional data analysis (FDA) provides tools for statistical modeling of functional data. Astronomical data presents several challenges for FDA methodology, e.g., sparse, irregular, and asynchronous sampling, and heteroscedastic measurement error. Bayesian FDA uses hierarchical Bayesian models for function populations, and is well suited to addressing these challenges. We provide an overview of astronomical functional data and some key Bayesian FDA modeling approaches, including functional mixed effects models, and stochastic process models. We briefly describe a Bayesian FDA framework combining FDA and machine learning methods to build low-dimensional parametric models for galaxy spectra.