The field of topological data analysis (TDA) came about after a series of papers by Gunnar Carlsson in 2005. Although the basic idea of persistent homology (the centerpiece of TDA) is very simple--applying the homology functor to a filtration of spaces--since then, a mathematical community has flourished. This talk aims to "sample" some of the mathematical contributions in applied topology. We will start at a familiar place: homotopy theory. We will discuss the Nerve Lemma as the "bread and butter" of TDA. Then, we will discuss the algebraic properties of persistence modules that make TDA useful. Next, we will shift our focus to generalizations of the basic construction: sublevel set filtrations, level set persistence, and Leray cosheaves (due to Justin Curry). Time permitting, I will talk about work by Andrew Blumberg and Mike Mandel that attempts to generalize persistent homology to a filtration of mapping spaces from a test space which, by the Yoneda lemma, are a complete invariant of the homotopy type of a filtration. There will be a lot of pictures throughout.