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The fields of topological data analysis (TDA) and geometric data analysis (GDA) use algebraic topology and differential geometry to capture topological and geometric structural properties of data that are not captured by other methods in data science and machine learning. The primary tool of TDA---and one of the focuses of this dissertation---is persistent homology, which measures the connected components, holes, and higher-dimensional voids of a data set and tracks how those voids emerge and disappear at different scales. The objective of GDA is to extract new insights by considering geometric invariants of a manifold, such as curvature, rather than topological invariants. Previous studies have demonstrated the power of geometry and topology for analyzing data in complex systems, neuroscience, biology, and many other fields.In my thesis, I study both the theory and applications of topological and geometric data analysis. In the first part of the dissertation, I establish and analyze a new construction, called a "persistence diagram (PD) bundle," for doing multiparameter TDA, and I develop an algorithm to compute a certain class of PD bundles. PD bundles generalize several important constructions in TDA: vineyards, the persistent homology transform, and fibered barcodes. In the second part of the dissertation, I apply TDA to several geospatial and geospatiotemporal data sets. In the last part of the dissertation, I introduce a new method for curvature estimation in point-cloud data.