Topological Data Analysis seeks to better understand and classify large data sets, including those in large dimensions, and extract useful topological properties to help analyze the data. Some applications of these techniques include medical image processing, machine learning, and signal analysis. One of these techniques is Persistent Homology, which attempts to recreate topological information by connecting points within a particular distance of each other. As this distance threshold increases, we can start to see which features are part of the overall "shape" of the data, and which might just be caused by noise in the data set. In this talk, we'll discuss the implementation of this method, look at some examples in low dimensions, and look at an application to the processing of images.