Explore topological modeling of complex data in this 55-minute AMS-MAA Invited Address from the 2018 Joint Mathematics Meetings. Delve into the challenges of big data analysis, comparing size versus complexity and examining traditional mathematical modeling approaches. Discover the limitations of algebraic modeling and clustering techniques before exploring the concept of data shape. Learn how to construct networks for data sets and apply topological modeling to various real-world scenarios, including diabetes analysis, breast cancer microarray studies, and microbiome research. Investigate feature modeling, hot spot analysis, and supervised analysis techniques. Gain insights into improving existing models, surface sub-populations in mortgage data, and leverage serendipity in exploratory data analysis.