A dataset can be represented in different mathematical forms; for example, a micrograph can be represented as an image, as a matrix, as a graph (network), or as an intensity function. These representations are used to perform transformations of the data with the goal of extracting different types of features such as spatial patterns, geometrical patterns, correlations, principal components, gradients of light, and frequencies. These features contain key information that facilitate visualization and analysis, detection of abnormalities, and construction of predictive models.  In this talk, we show how to use representations and transformations in innovative ways to analyze complex datasets arising in flow cytometry, liquid crystals, chemical processes, and molecular dynamics. We show how these tools can be used to design chemical sensors for the detection of contaminants in air and liquid mixtures, to predict reaction rates for acid-catalyzed reactions, to predict material properties from images, and to detect faults.