Topological data analysis provides machinery for summarizing the topology of complex data. The main goal of this project was to develop a new summary compatible with statistics and machine learning. This goal was met with the development of a new summary, the persistence landscape. This summary is stable, does not lose any information, has continuous and discrete versions, and obeys a strong law of large numbers and a central limit theorem. All of the standard tools in statistics and machine learning are available for subsequent analysis. For example, one can easily calculate averages and differences, apply principal component analysis and support vector machines, or feed these results into a neural network. The secondary goal of the project was to help place Topological Data Analysis on a firmer mathematical foundation, strengthening its connections to mathematics and making it easier for researchers to leverage mathematical results for analyzing complex data and complex networks. This goal has been met with the development a very general framework for topological data analysis. Both of these developments have led to much work by other researchers.