In recent years, technological innovations have facilitated the collection of complex, high-dimensional data that pose substantial modeling challenges. Most of the time, these complex objects are strongly characterized by internal structure that makes sparse representations possible. If we can learn a sparse set of features that accurately captures the salient features of a given object, then we can model these features using standard statistical tools including clustering, regression and classification. The key question is how well this sparse set of features captures the salient information in the objects. In this thesis, we develop methodology for evaluating latent feature representations for functional data and for using these latent features within functional regression frameworks to build flexible models. In the first project, we introduce a graphical latent feature representation tool (GLaRe) to learn features and assess how well a given feature learning approach captures the salient information in a data object. In the second project, we build on this feature learning methodology to propose a basis strategy for fitting functional regression models when the domain is a closed manifold. This methodology is applied to MRI data to characterize patterns of infant cortical thickness development in the first two years of life. In the third project, we adapt our feature learning and Bayesian functional regression methodology to high-frequency data streams. We model high-frequency intraocular pressure data streams using custom bases for quantile representations of the underlying distribution, and provide insights into the etiology of glaucoma.