In this talk I will describe a set of efficient computational methods for analyzing, quantifying and exploring relations and variability in structured data sets, such as collections of geometric shapes, point clouds, and large networks or graphs, among others. The main unifying theme of this work is the observation that many concepts in data analysis can be considered, both in theory and in practice, as linear operators acting on real-valued functions on the appropriate domains. Since real-valued functions can be defined on a wide variety of data representations and, as they enjoy a rich algebraic structure, such an approach has the potential to provide unified framework for representing and processing different types of data. I will give an overview of these and related techniques and demonstrate, in particular, how the functional operator point of view can be helpful in solving practical problems, including shape correspondence, tangent vector field processing and shape comparison among others.