There is a lot of discussion around the topic of “Big Data”, which means the study of large and complex data sets. From a mathematical point of view, these data sets typically have the structure of finite metric spaces, sometimes in numerous ways. The problem of analyzing these data sets is a very important one in the sciences, engineering, and commerce. Any kind of organizing principle for them would be a major contribution to the solution of these problems. It turns out that topology is a very useful such principle, both via versions of complex constructions as well as homology. This course will discuss these ideas, with numerous examples, and will as provide instruction on the use of persistent homology software.