Persistent homology is a powerful tool in topological data analysis for understanding the shape of data. High dimensional data, as found in deep learning and biology, presents novel challenges for standard techniques. We will explore what some of those challenges are, and why standard Vietoris-Rips constructions are insufficient to meet those challenges. Then we will explore some of the proposed solutions coming from different fields, and develop computationally tractable tools to allow us to leverage topological data analysis for high dimensional data.