Persistent homology is a widely used tool in Topological Data Analysis that encodes multi-scale topological information as a multi-set of points in the plane, called a persistence diagram. Each of these persistence points is associated with a lifetime (or persistence). Features with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological signal. We bring some statistical ideas to persistent homology in order to derive confidence sets that allow us to separate topological signal from topological noise. We also apply statistical theory to other topological descriptors such as the persistence landscape or silhouette, rather than working with the original diagrams or data sets. We motivate this work with three applications.